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Cyclotomic Hecke and Quiver Hecke Algebras

1 · Prerequisites

2 · Summary

Objective. This page develops the type-A cyclotomic Hecke algebras in the nondegenerate and degenerate branches, their charged finite-quantum-characteristic specializations, the central content sectors, and the Brundan-Kleshchev comparison with the type-A cyclotomic Khovanov-Lauda-Rouquier algebras. The first part organizes draft arguments for the universal affine and cyclotomic presentations: ordered-monomial bases, bounded spanning, the polynomial spectra that split over the residue values, canonical CRT content projectors, the level-one identification with the finite Hecke algebra, and the comparison maps between the Hecke and KLR presentations in both directions. The second part develops the arbitrary-level Murphy cellular theory with its Jucys-Murphy triangularity, the semistandard bases of permutation modules, the Schur algebra and the ordinary Schur functor, and the Jantzen/filtration theory: the DVR Gram valuation, the monic seminormal Gram determinants, their signed rim-hook valuation expansion, the Jantzen sum formula, and the resulting classification of blocks through Jantzen equivalence and content classes. The closing section addresses identification of the central content sectors with the blocks, the Brundan-Kleshchev blockwise isomorphism, and transport of the KLR grading to the Hecke blocks.

Conventions. The field is arbitrary and $e \geq 2$ is finite; the nondegenerate branch uses a Hecke parameter $q_{H} \neq 1$ of quantum characteristic $e$ , while the degenerate branch has $q_{H} = 1$ and $char F = e$ . The grading variable $v$ of the KLR side is independent of $q_{H}$ . The cyclotomic relation is imposed on the first polynomial generator and induction appends strands on the right. The e=0 ( $A_{\infty}$ ) branches are outside this page's cyclic scope and are recorded as a deferred boundary.

Reading order. The affine and cyclotomic presentations come first, then their bases and spectra, then the Hecke-KLR comparison; the Murphy/cellular and Schur theory follows the same dependency order, and the Jantzen and block theory depends on the trace and cell machinery. Items on this page are draft and are consumed by the examples companion and by the Ariki categorification pair.

Review status. The local affine bases, bounded cyclotomic spanning, CRT spectra and level-one identification have supported arguments. The Hecke–KLR comparison still requires the complete normalizing-series and colour-pattern calculations. The arbitrary-level basis and cellular chain also depend on unresolved cyclotomic KLR projectivity and commutator inputs. Trace perfectness, the degenerate-center lifting argument and the off-support signed rim-hook calculation remain open; their downstream Jantzen sum and primitive-block conclusions await those prerequisites.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules

Definition

Let $R$ be a commutative ring, let $n \geq 0$ and $ℓ \geq 1$ , and let $z, U_{1}, \dots, U_{ℓ}$ be units of $R$ (Algebras over a commutative ring, central structure maps, and algebra homomorphisms). The arbitrary-level cyclotomic Hecke algebra $H_{n} = H_{n} (R; z; U_{1}, \dots, U_{ℓ})$ is the $R$ -algebra presented by invertible commuting elements $X_{1}, \dots, X_{n}$ and generators $T_{1}, \dots, T_{n - 1}$ subject to the relations $(T_{i} - z) (T_{i} + 1) = 0, T_{i} T_{i + 1} T_{i} = T_{i + 1} T_{i} T_{i + 1}, T_{i} T_{j} = T_{j} T_{i} (∣ i - j ∣ > 1),$ $T_{i} X_{i} T_{i} = z X_{i + 1}, X_{j} T_{i} = T_{i} X_{j} (j \notin {i, i + 1}), \prod_{a = 1}^{ℓ} (X_{1} - U_{a}) = 0.$ For $n = 0$ set $H_{0} = R$ . The grading variable of this page is $v$ , not the unit $z$ : the two are identified only where a statement says so explicitly.

Multipartitions and Murphy data. An $ℓ$ -multipartition of $n$ is an ordered tuple $λ = (λ^{(1)}, \dots, λ^{(ℓ)})$ of partitions with $\sum_{b} ∣ λ^{(b)} ∣ = n$ (Tableaux and standard tableaux). Multipartitions are partially ordered by dominance, $λ ⊵ μ$ meaning that $\sum_{b < k} ∣ λ^{(b)} ∣ + \sum_{j \leq m} λ_{j}^{(k)} \geq \sum_{b < k} ∣ μ^{(b)} ∣ + \sum_{j \leq m} μ_{j}^{(k)}$ for all $k, m$ . Fill the diagram $t^{λ}$ with $1, \dots, n$ in component order, then row order, then column order. Put $a_{k} : = \sum_{b < k} ∣ λ^{(b)} ∣$ , $u_{λ} : = \prod_{k = 1}^{ℓ} \prod_{j \leq a_{k}} (X_{j} - U_{k})$ , $x_{λ} : = \sum_{w \in S_{λ}} T_{w}$ , and $m_{λ} : = x_{λ} u_{λ}$ , where $S_{λ}$ is the row subgroup: the subgroup of $S_{n}$ stabilizing the rows of $t^{λ}$ (The finite symmetric group $S_{n}$ , one-line notation, and cycle notation, The generic type-A Hecke algebra). Write $M_{λ} : = m_{λ} H_{n}$ for the right $H_{n}$ -submodule generated by $m_{λ}$ (Unital left and right modules over a ring; unqualified module means left module). The presentation involution $*$ of $H_{n}$ fixes every $X_{j}$ and every $T_{i}$ and reverses products. For standard tableaux $s, t$ of shape $λ$ put $m_{s t} : = T_{d (s)}^{*} m_{λ} T_{d (t)}$ , where the right tableau action is $t \cdot w : = w^{- 1} \cdot t$ (the left action relabels entries), and $d (s)$ is determined by $t^{λ} \cdot d (s) = s$ . Denote by $H^{> λ}$ the span of the higher Murphy shapes once their basis and cell law is established by The arbitrary-level cyclotomic Hecke Murphy basis is cellular ↗, and by $S_{λ}$ the right cell (Specht) module with basis $m_{t}$ and the action given by that cell law; its cellular form defines $D_{λ} : = S_{λ} / rad S_{λ}$ , which may vanish.

Types. The types used below are exactly all $ℓ$ -multipartitions of $n$ , including tuples with empty components — not all multicompositions. A semistandard tableau $S$ has entries the labels $(i, k)$ (row $i$ in component $k$ ) ordered by component first and then by row, weakly increasing along rows, strictly increasing down columns, and a label of component $a$ may occur only in physical component $b \leq a$ (Semistandard tableaux and Kostka numbers); the type of $S$ is the multipartition counting each label with the multiplicity of the corresponding row. For standard $s$ let $μ (s)$ be the tableau obtained from $s$ by replacing each entry by its row-and-component in $t^{μ}$ . Put $m_{(S, t)} : = \sum_{s : μ (s) = S} m_{s t}, m_{(S, T)} : = \sum_{s, t : μ (s) = S, ν (t) = T} m_{s t} .$ Let $M : = ⨁_{λ multipartition of n} M_{λ}, C_{n} : = {End}_{H_{n}} (M),$ with ordinary composition (The direct sum of an indexed family of modules). For $S, T$ of shape $λ$ and types $μ, ν$ , the map $φ_{S T}$ — once justified by Cyclotomic semistandard sums give the actual Schur endomorphism basis ↗ — sends $m_{α} h$ to $δ_{α, ν} m_{(S, T)} h$ . Its right cell module $Δ_{λ}$ has basis $v_{T}$ over the semistandard $T$ of multipartition types, with contravariant cell form and quotient $L_{λ}$ . Let $T^{λ}$ be the unique tableau of shape and type $λ$ , and let $e_{λ}$ be the identity projection onto $M_{λ}$ . Put $ω : = (\emptyset, \dots, \emptyset, (1^{n}))$ and $e : = e_{ω}$ ; then $m_{ω} = 1$ , so $M_{ω} = H_{n}$ , and $H_{n}$ is an actual direct summand of $M$ . The weight projections used by the Jantzen theory below are the $e_{ν}$ with $ν$ a multipartition. Write $K_{(λ, ν)} : = # {SStd}_{ν} (λ)$ for the number of semistandard tableaux of shape $λ$ and type $ν$ .

Boundary of the definition. All asserted module, form and map laws are discharged by the linked local suppliers The arbitrary-level cyclotomic Hecke Murphy basis is cellular ↗, Cyclotomic semistandard sums give the actual Schur endomorphism basis ↗ and Cyclotomic Schur standards and corner functor control block linkage ↗; the definition itself claims no proof of them and no primitive residue-content classification. No choice principle is used: only finite bases, finite subgroup and coset decompositions, finite filtrations and explicit polynomial identities occur.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Laurent binomial valuation at one over any field

Statement

Let $F$ be a field, let $F [t, t^{- 1}]$ be its Laurent polynomial ring (The Laurent polynomial ring as the principal localisation of Z[t] at t), and let $v_{1}$ be the $(t - 1)$ -adic valuation on the nonzero elements of $F [t, t^{- 1}]$ : for $0 \neq f$ , $v_{1} (f) : = \max {n \geq 0 : (t - 1)^{n} ∣ f in F [t, t^{- 1}]},$ so that $v_{1} (f) = 0$ exactly when $f (1) \neq 0$ (Discrete valuation rings). Let $A, B \in F ∖ {0}$ , let $u, v \in Z$ , put $N : = u - v$ and $f : = A t^{u} - B t^{v}$ .

If $A \neq B$ , then $v_{1} (f) = 0$ .
If $A = B$ and $N \neq 0$ , then $v_{1} (f) = 1$ when $char F = 0$ , and $v_{1} (f) = p^{v_{p} (∣ N ∣)}$ when $char F = p > 0$ ; here $v_{p}$ is the exponent of $p$ in $∣ N ∣$ , i.e. $N = p^{r} m$ with $p ∤ m$ (The characteristic of a ring: the least $n \geq 1$ with $n \cdot 1_{R} = 0$ when one exists, and $0$ otherwise).
If $A = B$ and $N = 0$ , then $f = 0$ , and the zero element has no finite valuation.

In the intended application the Laurent polynomials $Q_{a} q_{H}^{j - i} t^{n a + j - i}$ (foot residues of the Jantzen expression) are compared by this lemma: it computes their valuation exactly when their difference is nonzero and prevents replacing valuations by an equality-of-residues indicator in positive characteristic.

Facts & Assumptions

Given: A field $F$ , nonzero $A, B \in F$ , integers $u, v$ , the Laurent polynomial $f = A t^{u} - B t^{v}$ , and the $(t - 1)$ -adic valuation $v_{1}$ of the Statement.

[F1]

The finite Laurent construction over $F$ uses sums $\sum_{k} a_{k} t^{k}$ with finite support and the coefficientwise addition and convolution multiplication. It is the field-coefficient version of The Laurent polynomial ring as the principal localisation of Z[t] at t; $t$ is a unit, and multiplying by a sufficiently large power of $t$ identifies any such sum with an ordinary polynomial.

[F2]

A DVR is the ring of nonnegative values of a surjective integer-valued valuation on a field (Discrete valuation rings). The multiplicity valuation used here is established locally in step 1.1.

[F3]

The characteristic is the least positive integer annihilating $1_{F}$ , or zero if no such integer exists (The characteristic of a ring: the least $n \geq 1$ with $n \cdot 1_{R} = 0$ when one exists, and $0$ otherwise). Its field consequences needed here are established in step 1.1.

Proof

technique · direct

Given: The field $F$ , the Laurent polynomial ring $F [t, t^{- 1}]$ with unit $t$ , the valuation $v_{1}$ of the Statement, and the data of [F1]–[F3].

1.1F1F2F3algebra

Construct the valuation. For an ordinary nonzero polynomial $g$ , repeated division by $t - 1$ terminates as degree decreases, giving a unique factorization $g = (t - 1)^{a} g_{0}$ with $g_{0} (1) \neq 0$ ; division is explicit from $g (t) - g (1) = \sum_{j} a_{j} (t^{j} - 1)$ . For a Laurent polynomial multiply by $t^{M}$ first: since $t^{M} (1) = 1$ , the resulting multiplicity is independent of $M$ . Multiplying two factorizations multiplies their nonzero evaluations, so multiplicities add; adding them gives multiplicity at least their minimum, unless the sum is zero. Consequently $v (g / h) = v (g) - v (h)$ is a well-defined valuation on $F (t)$ , surjective since $v (t - 1) = 1$ . Its nonnegative-value ring is $F [t]_{(t - 1)}$ : cancel the powers of $t - 1$ in numerator and denominator, leaving a denominator nonzero at $1$ . This proves the DVR assertion in the sense of [F2], the zero-evaluation criterion, and invariance under Laurent units. Finally, positive field characteristic is prime: a composite least annihilator $a b$ would give $(a 1_{F}) (b 1_{F}) = 0$ with both factors nonzero. Division of an integer by that prime shows $m 1_{F} \neq 0$ if $p ∤ m$ .

1.2F1

The case $A = B$ , $N = 0$ . Then $f = A t^{u} - B t^{v} = A t^{u} - A t^{u} = 0$ by [F1], the zero element of $F [t, t^{- 1}]$ ; since $(t - 1)^{n} ∣ 0$ for every $n$ , the set in the definition of $v_{1}$ is unbounded and no finite value is assigned.

2.1F1F2step 1.1

Reduction to $N$ and to $A / B$ for $f \neq 0$ . Since $B t^{v}$ is a unit of $F [t, t^{- 1}]$ with inverse $B^{- 1} t^{- v}$ by [F1], the identity $A t^{u} - B t^{v} = B t^{v} ((A / B) t^{N} - 1)$ and [F2] give $v_{1} (A t^{u} - B t^{v}) = v_{1} ((A / B) t^{N} - 1)$ ; moreover if $N < 0$ then $t^{N} - 1 = - t^{N} (t^{- N} - 1)$ with $- t^{N}$ a unit, so $v_{1} (t^{N} - 1) = v_{1} (t^{∣ N ∣} - 1)$ by [F1] and [F2].

3.1F1F2step 1.1step 2.1

The case $A \neq B$ . Evaluating the Laurent polynomial $(A / B) t^{N} - 1$ at $t = 1$ gives $A / B - 1 \neq 0$ , so the factor $t - 1$ cannot divide $(A / B) t^{N} - 1$ : a product containing $t - 1$ would evaluate to $0$ under the ring homomorphism $f \mapsto f (1)$ with $1 \mapsto 1$ . Hence $v_{1} ((A / B) t^{N} - 1) = 0$ by [F2], and the reduction of step 2.1 gives $v_{1} (f) = 0$ .

3.2F1F2F3step 1.1step 2.1algebra

The case $A = B$ , $char F = 0$ , $N \neq 0$ . Here $f = A (t^{u} - t^{v}) = A t^{v} (t^{N} - 1)$ with $A t^{v}$ a unit, so by step 2.1 the valuation equals $v_{1} (t^{∣ N ∣} - 1)$ , while $t^{∣ N ∣} - 1 = (t - 1) (1 + t + \dots + t^{∣ N ∣ - 1})$ with second factor evaluating at $t = 1$ to $∣ N ∣ \cdot 1_{F} \neq 0$ by [F3] and therefore of valuation $0$ by [F2]; the valuation is additive on this factorisation of the nonzero polynomial $t^{∣ N ∣} - 1$ , so $v_{1} (f) = 1$ .

4.1F1F2F3step 1.1step 2.1algebra∎

The case $A = B$ , $char F = p > 0$ , $N \neq 0$ . Write $∣ N ∣ = p^{r} m$ with $r \geq 0$ and $p ∤ m$ . In characteristic $p$ the identity $(x - 1)^{p^{r}} = x^{p^{r}} - 1$ follows by iterating $(x - 1)^{p} = x^{p} - 1$ (the intermediate binomial coefficients $(\binom{p}{i})$ , $1 \leq i \leq p - 1$ , are multiples of $p$ ); applying it to $x = t^{m}$ gives $t^{∣ N ∣} - 1 = (t^{m} - 1)^{p^{r}} = (t - 1)^{p^{r}} (1 + t + \dots + t^{m - 1})^{p^{r}}$ . The second factor evaluates at $t = 1$ to $m^{p^{r}} \cdot 1_{F}$ , which is nonzero because $m \cdot 1_{F} \neq 0$ by [F3]; it thus has valuation $0$ by [F2], so $v_{1} (t^{∣ N ∣} - 1) = p^{r}$ , and step 2.1 gives $v_{1} (f) = p^{r} = p^{v_{p} (∣ N ∣)}$ .

Remarks

The lemma is a leaf computation over an arbitrary field: it is used where a denominator $t - 1$ is cleared or an integrality statement is checked at $t = 1$ , and it records explicitly that in characteristic $p$ the valuation is the full $p$ -power $p^{v_{p} (∣ N ∣)}$ , not merely the indicator of $p ∣ N$ . No choice principle is used.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Ultra-tight multisets admit integer column potentials

Statement

Let $r \geq 1$ and $L_{r} : = Z^{r} / Z (1, \dots, 1)$ (Free abelian group on a set, The integers as equivalence classes of pairs of naturals). For a finite multiset $X$ of elements of $L_{r}$ (The multiset construction $MSET (A)$ and the powerset construction $PSET (A)$ ), call $X$ tight if $(x_{k} - x_{l}) - (y_{k} - y_{l}) \leq 2$ for all occurrences $x, y \in X$ and all indices $k, l \in {1, \dots, r}$ . A zero exchange replaces two occurrences $x, y$ by $x - e_{k} + e_{l}$ and $y + e_{k} - e_{l}$ whenever $(x_{k} - x_{l}) - (y_{k} - y_{l}) = 2$ ; this relation is symmetric. Call $X$ ultra-tight if every finite zero-exchange sequence starting at $X$ ends in a tight multiset. Then $X$ is ultra-tight if and only if there is some $s \in Z^{r}$ with $x_{k} - x_{l} \leq s_{k} - s_{l} + 1 for every x \in X and every k, l .$ This includes the empty multiset and $r = 1$ .

Facts & Assumptions

Given: An integer $r \geq 1$ , the group $L_{r} = Z^{r} / Z (1, \dots, 1)$ , and a finite multiset $X$ of elements of $L_{r}$ .

[F1]

Every element of $L_{r}$ has a representative $(x_{1}, \dots, x_{r}) \in Z^{r}$ ; for $λ \in Z$ the representative $(x_{k} + λ)_{k}$ defines the same element, so the difference $x_{k} - x_{l}$ is well defined on $L_{r}$ , and negation $x \mapsto - x$ and the elements $x - e_{k} + e_{l}$ are well defined (Free abelian group on a set, The integers as equivalence classes of pairs of naturals).

[F2]

A finite multiset consists of finitely many occurrences with repetition; replacing a finite submultiset by another one of the same cardinality produces a multiset of the same cardinality, and any finite directed graph has a shortest directed path from a vertex $u$ to a vertex $v$ whenever a directed path from $u$ to $v$ exists (The multiset construction $MSET (A)$ and the powerset construction $PSET (A)$ ).

Proof

technique · direct

Given: The data of [F1]–[F2]: the group $L_{r}$ , a finite multiset $X$ in

$L_{r}$ , and the definitions of tight, zero exchange and ultra-tight of the

Statement.

1.1F1F2

Elementary cases and well-definedness. The empty multiset is ultra-tight (its images under zero-exchange sequences are empty, hence tight) and satisfies the displayed condition with $s = 0$ ; for $r = 1$ one has $L_{1} = Z / Z \cdot 1 = 0$ by [F1], so $x_{1} - x_{1} = 0$ for every occurrence, every finite multiset is tight, and no zero exchange exists because its defining equation reads $0 - 0 = 2$ ; thus every finite multiset in $L_{1}$ is ultra-tight and satisfies the condition with $s = 0$ . By [F1] the differences $x_{k} - x_{l}$ do not change when representatives are shifted by a multiple of $(1, \dots, 1)$ , so tightness and the zero-exchange steps are well defined on classes; a zero-exchange sequence may therefore be computed with any representatives.

1.2F1algebra

The reverse implication. Assume $s \in Z^{r}$ satisfies the displayed condition. Replacing each occurrence $x \in X$ by $x - s$ (a bijection of $L_{r}$ by [F1]) we may assume $x_{k} - x_{l} \leq 1$ for all $x, k, l$ ; applying this to the pair $(l, k)$ gives $x_{l} - x_{k} \leq 1$ , so $∣ x_{k} - x_{l} ∣ \leq 1$ for all $x, k, l$ . We prove by induction on the length of a zero-exchange sequence that every such multiset is tight, and suppress the shift by $s$ from the notation. Suppose the next exchange acts on $x, y \in X$ in coordinates $k, l$ , so $(x_{k} - x_{l}) - (y_{k} - y_{l}) = 2$ . Since $x_{k} - x_{l}, y_{k} - y_{l} \in [- 1, 1]$ , this forces $x_{k} - x_{l} = 1$ and $y_{k} - y_{l} = - 1$ ; writing $x^{'} = x - e_{k} + e_{l}$ and $y^{'} = y + e_{k} - e_{l}$ , the coordinate differences of $x^{'}$ are $x_{k}^{'} - x_{l}^{'} = - 1$ , and for $p \notin {k, l}$ we have $x_{k}^{'} - x_{p}^{'} = x_{k} - 1 - x_{p} \in [- 1, 1]$ , $x_{p}^{'} - x_{k}^{'} = x_{p} - x_{k} + 1 \in [- 1, 1]$ , $x_{l}^{'} - x_{p}^{'} = x_{k} - x_{p} \in [- 1, 1]$ and $x_{p}^{'} - x_{l}^{'} = x_{p} - x_{l} - 1 \in [- 1, 1]$ , using $x_{k} = x_{l} + 1$ and $∣ x_{p} - x_{q} ∣ \leq 1$ ; the same computation with $y$ in place of $x$ and $y_{l} = y_{k} + 1$ shows the same for $y^{'}$ . All other occurrences are unchanged and already have all coordinate differences in $[- 1, 1]$ . Hence in every reachable multiset all coordinate differences lie in $[- 1, 1]$ , and for any two occurrences $z, w$ of that multiset $(z_{k} - z_{l}) - (w_{k} - w_{l}) \leq 1 - (- 1) = 2$ : every reachable multiset is tight, so $X$ is ultra-tight.

2.1F1F2

Setup for the forward implication. Conversely assume $X$ ultra-tight; we prove by induction on $r$ that potentials $s \in Z^{r}$ exist satisfying the displayed condition. The case $r = 1$ is in step 1.1. Let $r \geq 2$ and assume the result for $r - 1$ . Let $π ⁣ : L_{r} \to L_{r - 1}$ forget the last coordinate; every zero exchange in $π (X)$ in coordinates $k, l \leq r - 1$ lifts to the zero exchange in $X$ with the same coordinates and the same labelled occurrences, because the defining difference of the pair is unchanged; hence every zero-exchange sequence in $π (X)$ is the projection of one in $X$ and ends tight. So $π (X)$ is ultra-tight, and by induction there are integers $s_{1}, \dots, s_{r - 1}$ with $x_{k} - x_{l} \leq s_{k} - s_{l} + 1$ for all $x \in X$ and $k, l \leq r - 1$ ; applying this to $(l, k)$ gives, with $a_{k l} : = s_{k} - s_{l}$ , $∣ (x_{k} - x_{l}) - a_{k l} ∣ \leq 1 (k, l \leq r - 1) .$ It remains to extend the potentials to the last coordinate: we must choose $s_{r}$ so that $∣ (x_{k} - x_{r}) - a_{k r} ∣ \leq 1$ for all $x \in X$ and all $k \leq r - 1$ , where now $a_{k r} = s_{k} - s_{r}$ . It is enough to do this in the descending sequence of stages described below: at stage $t = r - 1, \dots, 1$ one adds the bounds for the single pair $(t, r)$ while keeping all bounds already proved for pairs with both entries $\leq r - 1$ or both entries $\geq t + 1$ .

3.1F2step 2.1algebra

The stage $t = r - 1$ . Since $X$ is ultra-tight, the empty sequence shows that $X$ itself is tight; hence for all $x, y \in X$ , $(x_{r - 1} - x_{r}) - (y_{r - 1} - y_{r}) \leq 2$ and, exchanging $x$ and $y$ , $\geq - 2$ . The integers $x_{r - 1} - x_{r}$ therefore have range at most $2$ , so (choosing the integer midpoint) some integer $m$ satisfies $∣ (x_{r - 1} - x_{r}) - m ∣ \leq 1$ for all $x \in X$ ; choosing $s_{r} : = s_{r - 1} - m$ gives $a_{r - 1, r} = m$ and hence the pair- $(r - 1, r)$ bound, while all bounds of step 2.1 with $k, l \leq r - 1$ are unchanged.

4.1step 3.1given

Reduction to the exceptional case. Fix $t \leq r - 1$ and suppose the bounds $∣ (x_{k} - x_{l}) - a_{k l} ∣ \leq 1$ are known for all $x \in X$ whenever $k, l \leq r - 1$ or $k, l \geq t + 1$ (for $t = r - 1$ this is step 3.1 together with step 2.1). We show that the bounds also hold for all pairs with $k, l \geq t$ ; as observed in step 2.1, the only new pair to control is $(t, r)$ . If already $∣ (x_{t} - x_{r}) - a_{t r} ∣ \leq 1$ for all $x$ , there is nothing to prove. Otherwise some $x \in X$ has $x_{t} - x_{r} \geq a_{t r} + 2$ or $x_{t} - x_{r} \leq a_{t r} - 2$ ; replacing every occurrence $x$ by $- x$ and every $s_{k}$ by $- s_{k}$ (which preserves ultra-tightness and all current bounds by [F1]) we may assume the first case, so that for this $x$ , $x_{t} - x_{r} \geq a_{t r} + 2.$

5.1step 4.1algebra

Consequences of the choice of $x$ , and the sandwich. For every $u \in {t + 1, \dots, r - 1}$ the known bounds give $x_{t} - x_{u} \leq a_{t u} + 1$ and $x_{u} - x_{r} \leq a_{u r} + 1$ , so $x_{t} - x_{r} = (x_{t} - x_{u}) + (x_{u} - x_{r}) \leq a_{t u} + a_{u r} + 2 = a_{t r} + 2$ ; together with the choice of $x$ this forces equality throughout: $x_{t} - x_{u} = a_{t u} + 1, x_{u} - x_{r} = a_{u r} + 1 (t < u < r) .$ Tightness against $x$ gives $y_{t} - y_{r} \geq a_{t r}$ ; choosing any $t < u < r$ and adding the old upper bounds for $(t, u)$ and $(u, r)$ gives $y_{t} - y_{r} \leq a_{t r} + 2$ . Thus $a_{t r} \leq y_{t} - y_{r} \leq a_{t r} + 2 for every y \in X .$

6.1step 5.1algebra

The graph, the basic lower bound, and the case of no path. Draw a directed graph $G$ on ${1, \dots, r}$ with an arrow $k \to l$ if and only if some $y \in X$ satisfies $y_{k} - y_{l} = a_{k l} - 1$ . First we note that $x_{k} - x_{r} \geq a_{k r} + 1 for every k < r :$ for $k = u \geq t + 1$ this is the equality $x_{u} - x_{r} = a_{u r} + 1$ of step 5.1, for $k = t$ it follows from $x_{t} - x_{r} \geq a_{t r} + 2$ , and for $k < t$ it follows from $x_{k} - x_{r} = (x_{k} - x_{t}) + (x_{t} - x_{r}) \geq (a_{k t} - 1) + (a_{t r} + 2) = a_{k r} + 1$ using the known bound for the pair $(k, t)$ with $k, t \leq r - 1$ . Suppose first that $G$ contains no directed path from $t$ to $r$ . Let $T$ be the set of vertices reachable from $t$ (so $t \in T$ , $r \notin T$ ) and let $R$ be its complement; there is no arrow from $T$ to $R$ , since such an arrow would put its target in $T$ . Define new potentials ${\hat{s}}_{k} : = s_{k} + 1$ for $k \in T$ and ${\hat{s}}_{k} : = s_{k}$ for $k \in R$ , and set ${\hat{a}}_{k l} : = {\hat{s}}_{k} - {\hat{s}}_{l}$ . Then ${\hat{a}}_{t r} = a_{t r} + 1, {\hat{a}}_{k l} \in {a_{k l} - 1, a_{k l}, a_{k l} + 1},$ and ${\hat{a}}_{k l} = a_{k l} - 1$ occurs exactly for $k \in R$ , $l \in T$ . The new $(t, r)$ bound follows from the sandwich of step 5.1: ${\hat{a}}_{t r} - 1 = a_{t r} \leq y_{t} - y_{r} \leq a_{t r} + 2 = {\hat{a}}_{t r} + 1$ for all $y \in X$ . For any other pair the bound is immediate if ${\hat{a}}_{k l} \geq a_{k l}$ , since the old upper bound $y_{k} - y_{l} \leq a_{k l} + 1 \leq {\hat{a}}_{k l} + 1$ survives. If ${\hat{a}}_{k l} = a_{k l} - 1$ , then $k \in R$ , $l \in T$ ; the absence of an arrow from $T$ to $R$ says no $y$ has $y_{l} - y_{k} = a_{l k} - 1$ , so the old lower bound $y_{l} - y_{k} \geq a_{l k} - 1$ improves to $y_{l} - y_{k} \geq a_{l k}$ for all $y$ , i.e. $y_{k} - y_{l} \leq - a_{l k} = a_{k l} = {\hat{a}}_{k l} + 1$ . Thus the bounds hold for all pairs with both entries in ${t, \dots, r}$ , as required, and stage $t$ is complete in the no-path case.

7.1step 6.1given

The case of a path: the witnesses. Now suppose $G$ contains a directed path from $t$ to $r$ , and let $t = k_{1} \to k_{2} \to \dots \to k_{h} = r$ be one of minimal length (so $k_{1}, \dots, k_{h}$ are pairwise distinct by [F2]). For each $m = 1, \dots, h - 1$ choose an occurrence $y_{m} \in X$ witnessing the arrow $k_{m} \to k_{m + 1}$ , i.e. $y_{m, k_{m}} - y_{m, k_{m + 1}} = a_{k_{m} k_{m + 1}} - 1$ , Distinct edges have distinct witnessing occurrences. Indeed, for any witness put $z = y - s$ . The first $r - 1$ coordinates have spread at most one. A witness for two edges ending before $r$ would have the same low value at both starts and the same high value at both ends, giving a shortcut from the earlier start to the later end. For a last-edge witness, $z_{k_{h - 1}} = z_{r} - 1$ ; tightness against the chosen $x$ and step 6.1 give $z_{k} - z_{r} \geq - 1$ for $k < r$ , and the old spread bound gives $z_{k} - z_{r} \leq 0$ . If it also witnessed an earlier edge, that edge would start at $z_{r} - 1$ , giving a shortcut directly to $r$ . Both contradict shortestness. Now put $x^{(m)} : = x - e_{k_{m + 1}} + e_{r} (1 \leq m < h),$ so that $x^{(h - 1)} = x$ because $k_{h} = r$ . We shall prove by downward induction on $m$ that there is a multiset $Y_{m} \equiv X$ (in the equivalence relation generated by zero exchanges) containing $x^{(m)}$ and every occurrence among $y_{1}, \dots, y_{m}$ other than the chosen moving occurrence $x$ , and that the following two statements hold: $B_{m} : (x_{k_{m}}^{(m)} - x_{k_{m + 1}}^{(m)}) - (y_{m, k_{m}} - y_{m, k_{m + 1}}) = 2, C_{m} : x_{k_{m}} - x_{r} = a_{k_{m} r} + 1.$ Reading $C_{1}$ together with step 4.1 (which gives $x_{k_{1}} - x_{r} = x_{t} - x_{r} \geq a_{t r} + 2$ ) will then contradict the choice of $x$ ; the contradiction is completed below. We record $B_{h - 1}$ and $C_{h - 1}$ : by the basic lower bound of step 6.1, $(x_{k_{h - 1}} - x_{r}) - (y_{h - 1, k_{h - 1}} - y_{h - 1, r}) \geq (a_{k_{h - 1} r} + 1) - (a_{k_{h - 1} r} - 1) = 2$ , while tightness of $Y_{h - 1} : = X$ gives $\leq 2$ ; hence equality holds, which is $B_{h - 1}$ , and then $C_{h - 1}$ follows because $y_{h - 1, k_{h - 1}} - y_{h - 1, r} = a_{k_{h - 1} r} - 1$ and $(x_{k_{h - 1}} - x_{r}) - (y_{h - 1, k_{h - 1}} - y_{h - 1, r}) = 2$ .

8.1step 7.1algebra

The induction step constructing $Y_{m}$ . Let $m \leq h - 2$ and assume $A_{m + 1}$ and $B_{m + 1}$ , where $A_{m + 1}$ says that $Y_{m + 1} \equiv X$ contains $x^{(m + 1)}$ and every $y_{j}$ with $j \leq m + 1$ other than the original moving occurrence $x$ . Form $Y_{m}$ from $Y_{m + 1}$ by removing the occurrences $x^{(m + 1)}$ and $y_{m + 1}$ and adding the occurrences $x^{(m)}$ and $y_{m + 1} + e_{k_{m + 1}} - e_{k_{m + 2}}$ . By $B_{m + 1}$ the pair $(x^{(m + 1)}, y_{m + 1})$ has difference exactly $2$ in coordinates $(k_{m + 1}, k_{m + 2})$ , so this is a zero exchange; hence $Y_{m} \equiv Y_{m + 1} \equiv X$ by $A_{m + 1}$ , and the distinctness of the edge witnesses ensures that $Y_{m}$ retains every $y_{j}$ with $j \leq m$ other than the original moving occurrence $x$ , together with $x^{(m)}$ . Thus $A_{m}$ holds.

9.1step 7.1step 8.1algebra

The induction step forcing $B_{m}$ and $C_{m}$ , and the contradiction. Let $m \leq h - 2$ and assume $A_{m}$ and $C_{m + 1}$ . The witness $y_{m}$ cannot be the original moving occurrence $x$ : its edge equation together with $C_{m + 1}$ would give $x_{k_{m}} - x_{r} = (a_{k_{m} k_{m + 1}} - 1) + (a_{k_{m + 1} r} + 1) = a_{k_{m} r}$ , contradicting step 6.1. Thus $y_{m}$ is retained in $Y_{m}$ by $A_{m}$ . Since $k_{1}, \dots, k_{h}$ are pairwise distinct and only $k_{h}$ equals $r$ , the $r$ -th coordinate of $e_{r}$ vanishes in positions $k_{m}$ and $k_{m + 1}$ , so $x_{k_{m}}^{(m)} - x_{k_{m + 1}}^{(m)} = (x_{k_{m}} - x_{r}) - (x_{k_{m + 1}} - x_{r}) + 1 \geq (a_{k_{m} r} + 1) - (a_{k_{m + 1} r} + 1) + 1 = a_{k_{m} k_{m + 1}} + 1 = y_{m, k_{m}} - y_{m, k_{m + 1}} + 2,$ where the inequality uses the basic lower bound of step 6.1 for $k_{m}$ and statement $C_{m + 1}$ for $k_{m + 1}$ . Since $Y_{m}$ is tight (it lies in the class of the ultra-tight $X$ ), the reverse inequality holds and we have equality, which is $B_{m}$ ; reading the equality back through the displayed chain gives $x_{k_{m}} - x_{r} = a_{k_{m} r} + 1$ , which is $C_{m}$ . Downward induction on $m$ now gives $C_{1}$ , i.e. $x_{k_{1}} - x_{r} = x_{t} - x_{r} = a_{t r} + 1$ , contradicting $x_{t} - x_{r} \geq a_{t r} + 2$ from the choice of $x$ in step 4.1. Hence the exceptional case cannot occur: with the current potentials the pair $(t, r)$ is already bounded, so stage $t$ is complete in the path case as well.

10.1step 1.1step 6.1step 9.1∎

Completion of the induction. Stages $t = r - 1, r - 2, \dots, 1$ have now been treated: each stage either leaves the potentials unchanged, or (in the no-path case of step 6.1) replaces them by potentials for which the new bound holds, while all bounds proved earlier are preserved (the modification changes $s_{k} - s_{l}$ by at most $1$ in the necessary direction, and the argument of step 6.1 checks both the improved and the unchanged pairs). After stage $1$ the bounds hold for all pairs $k, l$ with $k, l \leq r - 1$ or $k, l \geq 1$ , i.e. for all $k, l \in {1, \dots, r}$ . This proves the forward implication for $r$ , and with the base case $r = 1$ of step 1.1 and the induction scheme of step 2.1, the forward implication for all $r \geq 1$ . Combined with the reverse implication of step 1.2, an ultra-tight multiset is exactly one admitting column potentials $s$ , including the empty multiset and the case $r = 1$ .

Remarks

The proof is a finite induction: it uses only finite multisets, integer arithmetic, finiteness of the directed graph and the existence of a shortest path among finitely many paths, so it uses no choice principle. The parametrization by $s \in Z^{r}$ is the one used by the arithmetic application on this page; Fayers' original parametrization by $a_{1}, \dots, a_{r - 1}$ is recovered by $a_{k l} = s_{k} - s_{l}$ with $s_{k} = \sum_{i \geq k} a_{i}$ .

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Affine type-A presentations and charged cyclotomic quotients

Definition

Let $A$ be a commutative ring and let $z \in A$ be a unit. Fix an integer $n \geq 0$ .

The nondegenerate affine presentation. The algebra $H_{n}^{a f f} (A; z)$ is the $A$ -algebra presented by commuting units $X_{1}, \dots, X_{n}$ and generators $T_{1}, \dots, T_{n - 1}$ (The generic type-A Hecke algebra, The quantum characteristic of an invertible Hecke parameter) subject to $T_{r}^{2} = (z - 1) T_{r} + z, T_{r} T_{r + 1} T_{r} = T_{r + 1} T_{r} T_{r + 1}, T_{r} T_{s} = T_{s} T_{r} (∣ r - s ∣ > 1),$ $T_{r} X_{r} T_{r} = z X_{r + 1}, T_{r} X_{s} = X_{s} T_{r} (s \notin {r, r + 1}),$ the last two relations being the Bernstein relation and the distant mixed commutation, and with $T_{r}$ invertible as a consequence of its quadratic relation. For $n = 0$ the algebra is $A$ .

The degenerate affine presentation. The algebra $H_{n}^{\deg} (A)$ is the $A$ -algebra presented by commuting generators $x_{1}, \dots, x_{n}$ and symmetric-group generators $s_{1}, \dots, s_{n - 1}$ subject to the Coxeter relations $s_{r}^{2} = 1$ , $s_{r} s_{r + 1} s_{r} = s_{r + 1} s_{r} s_{r + 1}$ , $s_{r} s_{t} = s_{t} s_{r}$ for $∣ r - t ∣ > 1$ together with the mixed relations $s_{r} x_{r} = x_{r + 1} s_{r} - 1, s_{r} x_{r + 1} = x_{r} s_{r} + 1, s_{r} x_{t} = x_{t} s_{r} (t \notin {r, r + 1}) .$

Auxiliary cyclotomic quotients. For a monic polynomial $f (u) = u^{ℓ} + a_{ℓ - 1} u^{ℓ - 1} + \dots + a_{0}$ in $A [u]$ of degree $ℓ$ impose the single relation $f (X_{1}) = 0$ (respectively $f (x_{1}) = 0$ ) on the FIRST polynomial generator and call the resulting quotient the monic cyclotomic quotient; when $ℓ = 0$ , so that $f = 1$ , the quotient at positive rank is the zero algebra, while rank zero remains $A$ . In the nondegenerate case the constant term $a_{0}$ is required to be a unit. The first-strand convention imposed here (cyclotomic relation on $X_{1}$ , and later induction appending on the right) is the one used on this page.

The charged cyclic specialization. Let $A = F$ be a field, let $z = q_{H} \in F^{\times}$ have finite quantum characteristic $e \geq 2$ (The quantum characteristic of an invertible Hecke parameter), and let $I = Z / e Z$ be the cyclic label set of residues, with a dominant cyclic type-A weight $(λ_{i})_{i \in I}$ , $λ_{i} \geq 0$ , and $ℓ = \sum_{i} λ_{i}$ (The cyclic affine type A Cartan datum and its residues). If $q_{H} \neq 1$ use the nondegenerate presentation and put $c_{i} : = q_{H}^{i}$ ; if $q_{H} = 1$ (which forces $char F = e$ ) use the degenerate presentation and put $c_{i} : = i \cdot 1_{F}$ . Impose $f (z_{1}) : = \prod_{i \in I} (z_{1} - c_{i})^{λ_{i}}, f (z_{1}) = 0.$ on the first generator ( $z_{1} = X_{1}$ in the nondegenerate branch and $z_{1} = x_{1}$ in the degenerate branch) and write $H_{d}^{Λ}$ for the resulting charged cyclotomic quotient of rank $d$ . This charged finite- $e$ branch is the scope of the present page; the arbitrary-base affine and monic definitions above supply its universal PBW and trace proof inputs. The definition itself asserts no basis, no spectrum, no primitive block decomposition and no Brundan–Kleshchev isomorphism. The grading variable $v$ of the KLR side is independent of $q_{H}$ .

Well-definedness notes. The nondegenerate parameters $c_{i} = q_{H}^{i}$ are distinct because $q_{H}$ has multiplicative order exactly $e$ ; the degenerate parameters $c_{i} = i \cdot 1_{F}$ are distinct because $char F = e$ makes $i \cdot 1_{F} \neq j \cdot 1_{F}$ for $0 \leq i < j < e$ . In the nondegenerate branch no embedding of the label set $Z / e Z$ into $F$ is claimed or used: $i$ is only a label for the value $q_{H}^{i}$ . All these definitions are given by generators, relations and a two-sided quotient ideal and use no choice principle.

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DVR cellular Jantzen filtration and exact Gram valuation

Statement

Let $O$ be a discrete valuation ring with uniformizer $π$ and residue field $F = O / π O$ (Discrete valuation rings), and let $ν_{π}$ denote its valuation. Let $M$ be a finite free $O$ -module with an $O$ -valued bilinear pairing $b$ (Unital left and right modules over a ring; unqualified module means left module) whose scalar extension to $Frac (O)$ is nondegenerate. For $s \geq 0$ put $M (s) : = {x \in M : b (x, M) \subseteq π^{s} O}, M_{F}^{s} : = (M (s) + π M) / π M .$ Then:

$M_{F}^{0} = M / π M$ and $M_{F}^{1} = rad (b_{F})$ , where $b_{F}$ is the reduced pairing on $M_{F}$ ;
the filtration terminates ( $M_{F}^{s} = 0$ for all sufficiently large $s$ ), and for any Gram matrix $G$ of $b$ one has the exact identity $\sum_{s > 0} \dim_{F} M_{F}^{s} = ν_{π} (\det G);$
if an $O$ -algebra $A$ acts on $M$ with an anti-involution $*$ satisfying $b (a x, y) = b (x, a^{*} y)$ for all $a \in A$ , $x, y \in M$ , then every $M (s)$ , and hence every $M_{F}^{s}$ , is an $A$ -submodule;
if $M_{F}$ is a cellular standard module and $b_{F}$ is its nonzero cellular form (Cellular bases and cell modules), then $rad (b_{F})$ is its unique maximal submodule and therefore coincides with its module radical (The radical, socle, head, and Loewy series of a finite-dimensional module).

The Smith reduction used in the proof works over $O$ without any characteristic restriction, in particular in characteristic $2$ ; it uses independent bases of $M$ and of $M^{*} = {Hom}_{O} (M, O)$ , never a diagonalization of a symmetric form.

Facts & Assumptions

Given: A discrete valuation ring $O$ with uniformizer $π$ , residue field $F$ , valuation $ν_{π}$ ; a finite free $O$ -module $M$ of rank $n$ with bilinear pairing $b$ , Gram matrix $G$ with respect to an $O$ -basis, and the submodules $M (s)$ of the Statement.

[F1]

Every nonzero element of $O$ is a unit times a unique power of $π$ , and $ν_{π}$ is additive on products; a square matrix over $O$ is invertible if and only if its determinant is a unit, and $ν_{π} (\det (U V)) = ν_{π} (\det U) + ν_{π} (\det V)$ (Discrete valuation rings, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

[F2]

$M^{*} = {Hom}_{O} (M, O)$ is free of rank $n$ , with dual basis $ε_{1}, \dots, ε_{n}$ ; an $O$ -linear map $φ ⁣ : M \to M^{*}$ has matrix $G^{T}$ in column coordinates, where $G = (b (u_{i}, u_{j}))$ , and $\det G \neq 0$ makes $φ$ injective with finitely generated torsion cokernel: the adjugate identity makes $\det G$ annihilate the cokernel, and the cokernel is generated by the images of a finite basis (Unital left and right modules over a ring; unqualified module means left module, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

[F3]

$M_{F} = M / π M$ carries the reduced pairing $b_{F} (\overset{ˉ}{x}, \overset{ˉ}{y}) : = b (x, y) m o d π$ ; $rad (b_{F}) = {x \in M_{F} : b_{F} (x, M_{F}) = 0}$ . The identification with the module radical in the cellular standard case is proved in step 1.5 (The radical, socle, head, and Loewy series of a finite-dimensional module).

[F4]

The fixed-first-index cell law and its starred fixed-last-index counterpart are the cellular axioms (Cellular bases and cell modules). Their rank-one form action is derived in step 1.5.

Proof

technique · direct

Given: The data of [F1]–[F4]: the DVR $O$ with uniformizer $π$ , the free

module $M$ with pairing $b$ and Gram matrix $G$ , the submodules $M (s)$ , and the

reduced pairing $b_{F}$ .

1.1givenF1algebra

The rank-zero case. If $n = 0$ , then $M = 0$ and $M^{*} = 0$ , the pairing is generically nondegenerate on the zero space, and $M (s) = M_{F}^{s} = 0$ for every $s$ . The empty Gram determinant is defined here by the empty Leibniz product to be $1$ , whose valuation is $0$ ; the empty sum of elementary-divisor exponents is $0$ . Thus the Gram valuation identity is $0 = 0$ , the radical and stability assertions hold, and the conditional nonzero-cellular-form assertion has no rank-zero instance. For the remaining steps assume $n \geq 1$ .

1.2F1F2construct

Smith reduction of the form. View $b$ as the $O$ -linear map $φ ⁣ : M \to M^{*}$ , $φ (x) = b (x, -)$ , with matrix $G^{T}$ in column coordinates by [F2]; nondegeneracy over $Frac (O)$ gives $\det G \neq 0$ , so $φ$ has full rank. Choose a nonzero matrix entry of $φ$ of least valuation $ν_{π}$ and move it to position $(1, 1)$ by permuting the bases, then multiply its row by a unit of $O$ to make it $π^{a_{1}}$ ; every entry of the matrix is divisible by $π^{a_{1}}$ by minimality (including zero entries). Subtracting suitable $O$ -multiples of the first row from the other rows and of the first column from the other columns (all coefficients in $O$ , all operations unimodular by [F1]) kills the first row and column outside position $(1, 1)$ , leaving a smaller matrix over $O$ . Iterating over the remaining rows and columns produces bases $u_{1}, \dots, u_{n}$ of $M$ and $v_{1}, \dots, v_{n}$ of $M^{*}$ with $φ (u_{j}) = π^{a_{j}} v_{j}$ and $0 \leq a_{1} \leq \dots \leq a_{n}$ .

1.3F3algebra

Reduction of the radical. For $\overset{ˉ}{x} \in M_{F}$ one has $b_{F} (\overset{ˉ}{x}, M_{F}) = 0$ if and only if $b (x, M) \subseteq π O$ , i.e. $x \in M (1)$ ; hence $rad (b_{F}) = (M (1) + π M) / π M = M_{F}^{1}$ by [F3], the containment $π M \subseteq M (1)$ being immediate from $b (π x, M) \subseteq π O$ .

1.4F3algebra

Submodule stability. Let $a \in A$ and $x \in M (s)$ ; by the hypothesis $b (a x, y) = b (x, a^{*} y) \in π^{s} O$ for every $y \in M$ , since $a^{*} y \in M$ . Hence $a x \in M (s)$ and $M (s)$ is an $A$ -submodule; consequently $M_{F}^{s} = (M (s) + π M) / π M$ is an $A$ -submodule of $M_{F}$ , the $π M$ -cosets of $M (s)$ being stable because $π M$ is stable.

1.5F3F4algebra

The cellular standard module. In a cell layer, applying both axioms [F4] to $c_{u s} c_{t v}$ shows that the product is $b_{F} (s, t) c_{u v}$ modulo higher shapes: right multiplication retains $u$ , left multiplication retains $v$ , and comparison makes the scalar independent of $u, v$ . This is the cellular form; the cell action therefore sends a vector $w$ to $b_{F} (v, w) u$ in the left convention, with indices transposed in the right convention. Assume $M_{F}$ is a cellular standard module and $b_{F}$ is its nonzero cellular form, and let $N \subseteq M_{F}$ be a proper submodule. If some $w \in N$ lies outside $rad (b_{F})$ , choose a label $v$ with $b_{F} (v, w) \neq 0$ ; by the rank-one action [F4] every basis vector $u$ equals $b_{F} (v, w)^{- 1}$ times the image of $w$ under the cell element of the pair $(u, v)$ , so $u \in N$ for every basis label $u$ and $N = M_{F}$ , a contradiction. Thus every proper submodule is contained in $rad (b_{F})$ , which is proper because the form is nonzero, and the same rank-one argument shows that the quotient $M_{F} / rad (b_{F})$ is simple (it has no nonzero proper submodule). Hence $rad (b_{F})$ is the unique maximal submodule of $M_{F}$ , i.e. its module radical by [F3].

2.1step 1.2algebra

The filtration in the Smith bases. In the bases of step 1.2 one computes for $s \geq 0$ that $b (u_{j}, M) = π^{a_{j}} O$ , so $M (s) = ⨁_{j = 1}^{n} π^{\max (0, s - a_{j})} O u_{j}, \dim_{F} M_{F}^{s} = # {j : a_{j} \geq s} .$ In particular $M (0) = M$ , so $M_{F}^{0} = M / π M$ ; the reduced filtration terminates because every summand of $M (s)$ is contained in $π M$ for $s > \max_{j} a_{j}$ , so $M_{F}^{s} = 0$ ; and each $M_{F}^{s}$ is a subspace of the finite-dimensional $F$ -space $M_{F}$ .

3.1F1F2step 2.1algebra

The Gram valuation identity. By the matrix form of step 1.2, the determinant of $G$ equals that of $G^{T}$ and hence $\det (π^{a_{j}} δ_{j k})$ times a unit, since both change-of-basis matrices are invertible over $O$ ; hence, using additivity of $ν_{π}$ by [F1], $ν_{π} (\det G) = \sum_{j = 1}^{n} a_{j} = \sum_{s > 0} # {j : a_{j} \geq s} = \sum_{s > 0} \dim_{F} M_{F}^{s},$ the middle equality counting each $a_{j}$ as the number of positive integers $s \leq a_{j}$ .

4.1step 1.1step 2.1step 3.1step 1.3step 1.4step 1.5∎

Conclusion. The rank-zero argument of step 1.1 and steps 1.3–1.5 and 2.1–3.1 establish all four assertions of the Statement: the identification of $M_{F}^{0}$ and $M_{F}^{1} = rad (b_{F})$ , the termination together with the exact identity $\sum_{s > 0} \dim_{F} M_{F}^{s} = ν_{π} (\det G)$ , the submodule stability under an anti-involution action, and the equality of $rad (b_{F})$ with the module radical in the cellular standard case. The Smith reduction of step 1.2 is valid over any discrete valuation ring, including residue characteristic $2$ , because it uses independent domain and codomain bases and no symmetric diagonalization.

Remarks

The exactness of the sum formula is the point of the lemma: the multiplicities $\dim_{F} M_{F}^{s}$ of the Jantzen filtration are read off from the $π$ -adic valuation of the Gram determinant, and in positive residue characteristic the valuation is the full $p$ -power of the corresponding binomial coefficient, not merely an indicator. No choice principle is used: all bases are obtained by finitely many unimodular operations on finite matrices.

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Universal nondegenerate affine Hecke ordered monomials form a basis

Statement

Let $R : = Z [z^{\pm 1}, U_{1}^{\pm 1}, \dots, U_{ℓ}^{\pm 1}]$ be the universal Laurent parameter ring of Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules, and let $H_{n}^{a f f}$ be the $R$ -algebra presented there by the commuting units $X_{1}, \dots, X_{n}$ , the generators $T_{1}, \dots, T_{n - 1}$ and the quadratic, braid, distant, Bernstein and mixed-commutation relations — without imposing the first-variable cyclotomic relation. Then:

both families $X^{a} T_{w} and T_{w} X^{a} (a \in Z^{n}, w \in S_{n})$ are $R$ -bases of $H_{n}^{a f f}$ , and the same two families are bases after arbitrary base change to a commutative ring in which $z$ is a unit;
left multiplication by any nonzero element of the Laurent polynomial subalgebra $R [X_{1}^{\pm 1}, \dots, X_{n}^{\pm 1}] \subseteq H_{n}^{a f f}$ is injective;
after localizing at the differences $X_{i} - X_{j}$ ( $i \neq j$ ), the algebra maps into the rational crossed product of the fraction field of the polynomial ring with $S_{n}$ , and in that localization the elements $Φ_{i} : = T_{i} - \frac{(z - 1) X_{i + 1}}{X_{i + 1} - X_{i}} = T_{i} - c_{i, i + 1}, c_{i j} : = \frac{(z - 1) X_{j}}{X_{j} - X_{i}},$ satisfy $Φ_{i} f = s_{i} (f) Φ_{i}$ for every (localized) Laurent polynomial $f$ .

No cyclotomic basis, Brundan–Kleshchev theorem, trace form or primitive-content assertion is used.

Facts & Assumptions

Given: The universal Laurent parameter ring $R$ , the affine presentation $H_{n}^{a f f}$ of the Statement, the symmetric group $S_{n}$ with its Coxeter presentation, and the parameters $c_{i j} = (z - 1) X_{j} / (X_{j} - X_{i})$ and $d_{i j} = (X_{j} - z X_{i}) / (X_{j} - X_{i})$ in the rational crossed product.

[F1]

In $H_{n}^{a f f}$ the relations $(T_{i} - z) (T_{i} + 1) = 0$ , $T_{i} T_{i + 1} T_{i} = T_{i + 1} T_{i} T_{i + 1}$ , $T_{i} T_{j} = T_{j} T_{i}$ for $∣ i - j ∣ > 1$ , $T_{i} X_{i} T_{i} = z X_{i + 1}$ and $X_{j} T_{i} = T_{i} X_{j}$ for $j \notin {i, i + 1}$ hold, and $X_{1}, \dots, X_{n}$ are commuting units (Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules).

[F2]

The quadratic relation makes $T_{i}$ invertible, with $T_{i}^{- 1} = z^{- 1} (T_{i} - (z - 1))$ ; the $T_{i}$ generate a quotient of the generic type-A Hecke algebra, whose standard elements $T_{w}$ ( $w \in S_{n}$ ) span the subalgebra they generate; their independence in the affine algebra is proved below (The standard basis of the generic type-A Hecke algebra).

[F3]

$S_{n}$ has the adjacent-transposition Coxeter presentation (The symmetric group has the Coxeter presentation). Set $ℓ (w)$ to be the minimum number of adjacent transpositions expressing $w$ ; a reduced expression means one attaining this minimum. The subword length estimate needed below is proved directly in step 1.3.

Proof

technique · direct

Given: The universal domain $R$ , its Laurent polynomial ring $P = R [X_{1}^{\pm 1}, \dots, X_{n}^{\pm 1}]$ , the fraction field $K = Frac (P)$ , and the data of [F1]–[F3].

1.1F1F2algebra

Ordered spanning without localization. For every $f \in P$ the presentation relations give $T_{i} f = s_{i} (f) T_{i} + c_{i, i + 1} (f - s_{i} (f))$ , where $c_{i, i + 1} = (z - 1) X_{i + 1} / (X_{i + 1} - X_{i})$ . The correction is in $P$ : $f - s_{i} (f)$ is divisible by $X_{i + 1} - X_{i}$ , first for ordinary monomials by the difference-of-powers identity, and then for Laurent monomials after multiplying by a symmetric Laurent monomial unit. For $f = X_{i}$ this is the Bernstein relation, for $f = X_{i + 1}$ it is its conjugate using the quadratic relation, and the general formula follows by addition and the product rule $f g - s_{i} (f g) = (f - s_{i} (f)) g + s_{i} (f) (g - s_{i} (g))$ , including inverse generators by applying that rule to $f f^{- 1} = 1$ . Moving a polynomial across a $T$ reduces the number of misplaced polynomial factors, and correction terms reduce the number of $T$ factors; the finite Hecke relations then span the remaining words by $T_{w}$ . Thus $H_{n}^{a f f}$ is spanned over $P$ by the $T_{w}$ , and the presentation anti-involution fixing $X_{i}, T_{i}$ gives right ordered spanning too.

1.2F1algebra

A representation of the unlocalized presentation. Form $K ⋊ S_{n}$ with multiplication $(f σ) (g τ) = f σ (g) σ τ$ . For distinct indices set $c_{i j} = (z - 1) X_{j} / (X_{j} - X_{i})$ and $d_{i j} = (X_{j} - z X_{i}) / (X_{j} - X_{i}) = z - c_{i j}$ , and assign $T_{i} \mapsto c_{i, i + 1} + d_{i, i + 1} s_{i}$ , $X_{j} \mapsto X_{j}$ . The rational identities $c_{i j} + c_{j i} = z - 1$ and $d_{i j} d_{j i} = z + (z - 1) c_{i j} - c_{i j}^{2}$ give the quadratic relation. For three distinct indices, expansion of the two braid products in their six permutation coefficients agrees: the longest coefficients are $d_{i j} d_{i k} d_{j k}$ in either order; the two length-two coefficients agree by commuting scalar factors; the length-one and identity coefficients agree using $c_{i j} c_{j k} = c_{i k} (c_{i j} + c_{j k} - (z - 1))$ together with the preceding quadratic identities. These identities follow by multiplying by the common product of the differences and expanding the polynomial numerators. Distant relations follow from disjoint variable pairs. Finally $T_{i} f - s_{i} (f) T_{i} = c_{i, i + 1} (f - s_{i} (f))$ , so the Bernstein and mixed relations hold. All denominators here are nonzero polynomials in independent variables in the domain $P$ . This defines a homomorphism from the unlocalized presented algebra to $K ⋊ S_{n}$ .

1.3F3step 1.1step 1.2algebra

Independence over the base ring. For a reduced word $w = s_{i_{1}} \dots s_{i_{k}}$ , the coefficient of the group element $w$ in the image of $T_{w}$ is $D_{w} = d_{i_{1}, i_{1} + 1} s_{i_{1}} (d_{i_{2}, i_{2} + 1}) \dots (s_{i_{1}} \dots s_{i_{k - 1}}) (d_{i_{k}, i_{k} + 1}) \neq 0.$ Every factor is a nonzero rational function. Every other term uses at most $k - 1$ of the $k$ adjacent transpositions, so its permutation has minimum word length at most $k - 1$ , directly by the definition of length in [F3]. No Bruhat-order assertion is needed. In a finite relation $\sum_{w} f_{w} T_{w} = 0$ with $f_{w} \in P$ , select a label $w$ of maximal length with $f_{w} \neq 0$ . Its group-element coefficient is $f_{w} D_{w}$ and receives no contribution from any other label of that maximal length or from shorter labels, so it is nonzero in $K$ , a contradiction. Hence the $T_{w}$ are $P$ -linearly independent and the representation is injective. Since the Laurent monomials $X^{a}$ are an $R$ -basis of $P$ , this proves that $X^{a} T_{w}$ are an $R$ -basis; the anti-involution proves the right basis. Tensoring this free basis with any commutative $R$ -algebra preserves it, and tensoring the generators-and-relations presentation gives exactly the same affine presentation over that algebra. As the unused $U$ parameters do not enter the relations, any commutative coefficient ring with unit $z$ is obtained by sending every $U_{a}$ to $1$ . Thus the asserted arbitrary base change is valid, without inverting differences in the base-changed algebra.

2.1step 1.1step 1.2step 1.3algebra

Localization and intertwiners. Let $S$ be the multiplicative set generated by all $X_{i} - X_{j}$ with $i \neq j$ . It is invariant under $S_{n}$ . The mixed formula of step 1.1 extends to $S^{- 1} P$ by the same product rule and by solving the formula for inverses of elements of $S$ ; the correction remains in $S^{- 1} P$ . Consequently adjoining these inverses gives the algebra with free left coefficient module $⨁_{w} (S^{- 1} P) T_{w}$ and the same multiplication rules. This realizes the localization: the inverse formulas make it universal among homomorphisms inverting $S$ , and its injective representation in $K ⋊ S_{n}$ follows from the identical maximal-length argument of step 1.3. In it put $Φ_{i} = T_{i} - c_{i, i + 1}$ ; the mixed formula gives $Φ_{i} f = s_{i} (f) Φ_{i}$ for all $f \in S^{- 1} P$ . Products $Φ_{w}$ along reduced words have unitriangular expansions in the $T_{v}$ , because selecting every $T$ gives $T_{w}$ and selecting any scalar correction leaves strictly shorter words; thus they too form a basis over $S^{- 1} P$ . No parameter difference $U_{a} - U_{b}$ , quantum factorial or Gram determinant is inverted.

2.2step 1.3algebra

Polynomial multiplication is injective over the universal domain. If $0 \neq f \in P$ and $h = \sum_{w} f_{w} T_{w}$ , then $f h = 0$ gives $f f_{w} = 0$ for each $w$ by step 1.3. Since $P$ is a domain, every $f_{w} = 0$ , so $h = 0$ . The right-basis argument gives the analogous right multiplication result. This injectivity assertion is over the universal domain, as in Statement (2), and does not assert injectivity of nonzero zero divisors after arbitrary base change.

3.1F3step 2.1algebra

Subword support in the localization. Expanding a reduced product $T_{w} = \prod_{t} (Φ_{i_{t}} + c_{i_{t}, i_{t} + 1})$ and moving coefficients left using step 2.1 gives terms supported on products of subwords of the reduced word. Every coefficient is a sum of products of permuted $c$ functions and hence has only variable-difference denominators. Equivalently, in the crossed product $Φ_{i} = d_{i, i + 1} s_{i}$ , so moving a localized polynomial $f$ through such a term replaces it by the corresponding subword permutation applied to $f$ . This proves the subword expansion used by consumers and the asserted intertwining throughout the embedded localization.

4.1step 1.3step 2.2step 2.1step 3.1algebra∎

Conclusion. Step 1.3 proves both ordered bases over $R$ and after arbitrary base change with unit $z$ ; step 2.2 proves the universal-domain injectivity; and steps 2.1 and 3.1 prove the localized embedding and intertwining identity. For $n = 0$ there are no polynomial or Hecke generators, $S_{0}$ consists of the identity, the lone ordered monomial is $1$ , and $H_{0}^{a f f} = R$ ; all localization and intertwining clauses are then vacuous. No cyclotomic basis, trace form, primitive-content assertion or Brundan–Kleshchev result is used.

Remarks

The universal statement is stated over the characteristic-zero Laurent domain and then specializes by base change, so all polynomial identities in the proof are identities in $R$ . The proof uses no choice principle: all straightening, triangularity and expansion steps are finite.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Bounded ordered monomials span the cyclotomic Hecke quotients

Statement

Over any commutative base ring $A$ with invertible nondegenerate parameter, use the defining relations of Affine type-A presentations and charged cyclotomic quotients, writing $q_{H}$ for the nondegenerate Hecke parameter. For a monic cyclotomic polynomial $f$ of degree $ℓ > 0$ , with unit constant term in the nondegenerate branch, the quotient by $f (X_{1}) = 0$ is spanned by $X_{1}^{a_{1}} \dots X_{d}^{a_{d}} T_{w}, 0 \leq a_{r} < ℓ, w \in S_{d},$ and the degenerate quotient by $f (x_{1}) = 0$ is spanned by $x_{1}^{a_{1}} \dots x_{d}^{a_{d}} w, 0 \leq a_{r} < ℓ, w \in S_{d} .$ In particular, over a field $F$ each quotient has dimension at most $ℓ^{d} d!$ . This applies to $f (t) = \prod_{i \in I} (t - q_{H}^{i})^{λ_{i}}$ and $f (t) = \prod_{i \in I} (t - i)^{λ_{i}}$ in the two finite-quantum-characteristic branches, with $ℓ = \sum_{i} λ_{i}$ . No independence and no separate relation $f (X_{r}) = 0$ or $f (x_{r}) = 0$ for $r > 1$ is asserted. If $ℓ = 0$ and $d > 0$ , the imposed relation $1 = 0$ gives the zero algebra with empty spanning family; for $d = 0$ the algebra is $A$ with the singleton empty monomial.

Facts & Assumptions

Given: The two presentations of Affine type-A presentations and charged cyclotomic quotients over $A$ , a monic cyclotomic polynomial $f$ of degree $ℓ > 0$ , and the quotients of the Statement.

[F1]

In the nondegenerate presentation the mixed relation is $T_{r} X_{r} T_{r} = q_{H} X_{r + 1}$ with $q_{H}$ invertible, $T_{r}^{2} = (q_{H} - 1) T_{r} + q_{H}$ , the distant and braid relations hold, and $T_{r}$ is invertible with $T_{r}^{- 1} = q_{H}^{- 1} (T_{r} - q_{H} + 1)$ ; in the degenerate presentation $s_{r} x_{r} = x_{r + 1} s_{r} - 1$ , $s_{r} x_{r + 1} = x_{r} s_{r} + 1$ , $s_{r} x_{t} = x_{t} s_{r}$ for $t \notin {r, r + 1}$ , and the Coxeter relations hold (Affine type-A presentations and charged cyclotomic quotients, The generic type-A Hecke algebra).

[F2]

The $T_{r}$ generate a quotient of the generic type-A Hecke algebra with standard basis ${T_{w} : w \in S_{d}}$ , and the $s_{r}$ generate the quotient of $A [S_{d}]$ with basis $S_{d}$ ; finite straightening in the Coxeter group reduces each word in the $T_{r}$ (respectively $s_{r}$ ) to a linear combination of the standard words (The standard basis of the generic type-A Hecke algebra, The symmetric group has the Coxeter presentation).

Proof

technique · direct

Given: The two presentations over $A$ , the cyclotomic quotient by

$f (X_{1}) = 0$ (nondegenerate) or $f (x_{1}) = 0$ (degenerate), and the data of

[F1]–[F2].

1.1F1F2algebra

Elementary straightening of polynomials past a simple generator. In the nondegenerate branch the mixed relation gives, for a polynomial $p$ in the commuting variables, $T_{r} p = (s_{r} p) T_{r} - (q_{H} - 1) X_{r + 1} \frac{p - s_{r} p}{X_{r} - X_{r + 1}},$ and in the degenerate branch $s_{r} p = (s_{r} p) s_{r} - \frac{p - s_{r} p}{x_{r} - x_{r + 1}} .$ Both divided differences are polynomials: for $p = X_{r}^{m}$ the first identity is $T_{r} X_{r}^{m} = X_{r + 1}^{m} T_{r} - (q_{H} - 1) (X_{r}^{m - 1} + X_{r}^{m - 2} X_{r + 1} + \dots + X_{r + 1}^{m - 1}) X_{r + 1}$ , which follows from [F1] by induction on $m$ , and the second is $s_{r} x_{r}^{m} = x_{r + 1}^{m} s_{r} - \sum_{k = 0}^{m - 1} x_{r}^{k} x_{r + 1}^{m - 1 - k}$ , again by induction; the general case follows from the divided-difference product rule for the two variable replacements $X_{r} \leftrightarrow X_{r + 1}$ and $x_{r} \leftrightarrow x_{r + 1}$ . Consequently any word in the generators and polynomial variables rewrites as an $A$ -linear combination of ordered monomials $X^{a} T_{w}$ in the nondegenerate branch (the $T$ -word reduced by [F2], polynomial factors collected by commutation) and of $x^{a} w$ in the degenerate branch. The nondegenerate straightening preserves total polynomial degree; the degenerate one lowers degree by one in the correction term.

2.1F1step 1.1algebra

The nondegenerate quotient is generated by positive powers. In the nondegenerate branch the relation $f (X_{1}) = 0$ has nonzero constant term $a_{0}$ , so $X_{1}^{- 1} = - a_{0}^{- 1} (X_{1}^{ℓ - 1} + a_{ℓ - 1} X_{1}^{ℓ - 2} + \dots + a_{1})$ is a polynomial in $X_{1}$ ; and from $T_{r} X_{r} T_{r} = q_{H} X_{r + 1}$ one gets $X_{r + 1}^{- 1} = q_{H} T_{r}^{- 1} X_{r}^{- 1} T_{r}^{- 1}$ by inverting the equation. Starting from the displayed polynomial expression for $X_{1}^{- 1}$ and iterating this forward recursion expresses every $X_{r}^{- 1}$ through positive powers of $X_{1}$ and the $T$ 's, since $T_{r}^{- 1} = q_{H}^{- 1} (T_{r} - q_{H} + 1)$ is a polynomial in $T_{r}$ . Therefore the quotient is already generated, as an $A$ -module, by words in $X_{1}, \dots, X_{d}$ and the $T_{r}$ , and step 1.1 applies to them.

2.2F1F2step 1.1algebra

The transported relations in the nondegenerate branch. Put $R_{1} : = f (X_{1})$ and $R_{r + 1} : = T_{r} R_{r} T_{r}$ ; each $R_{r}$ is zero in the quotient. We claim by induction on $r$ that ordered straightening writes $R_{r} = q_{H}^{r - 1} X_{r}^{ℓ} + \sum_{a, w} c_{a, w} X_{1}^{a_{1}} \dots X_{r}^{a_{r}} T_{w},$ where $w \in S_{r}$ , $\sum_{j} a_{j} \leq ℓ$ , $a_{r} < ℓ$ , and no variable of index above $r$ occurs. The case $r = 1$ is the definition, with constant coefficient rearranged. For the induction step, direct summation of the mixed relation gives, for $m \geq 1$ , $T_{r} X_{r}^{m} T_{r} = q_{H} X_{r + 1}^{m} - (q_{H} - 1) \sum_{k = 1}^{m - 1} X_{r}^{k} X_{r + 1}^{m - k} T_{r},$ and for $m = 0$ one uses $T_{r}^{2} = (q_{H} - 1) T_{r} + q_{H}$ . Transporting the leading term $q_{H}^{r - 1} X_{r}^{ℓ}$ therefore produces exactly $q_{H}^{r} X_{r + 1}^{ℓ}$ plus terms of $X_{r + 1}$ -degree below $ℓ$ . Each remainder monomial has $X_{r}$ -degree $a_{r} < ℓ$ ; variables of index below $r$ commute with $T_{r}$ and with $X_{r + 1}$ , and $T_{w} \in H_{r}$ for $w \in S_{r}$ , so applying the displayed formula to $X_{r}^{a_{r}}$ and then straightening the finite words $T_{r} T_{w} T_{r}$ , $T_{w} T_{r}$ by [F2] produces terms of $X_{r + 1}$ -degree at most $a_{r} < ℓ$ and total polynomial degree at most $ℓ$ . This proves the claim.

2.3F1F2step 1.1algebra

The transported relations in the degenerate branch. Put $R_{1} : = f (x_{1})$ and $R_{r + 1} : = s_{r} R_{r} s_{r}$ . Straightening and induction on $r$ show $R_{r} = x_{r}^{ℓ} + \sum_{a, w} c_{a, w} x^{a} w,$ where $w \in S_{r}$ , all occurring variables have index at most $r$ , and $\sum_{j} a_{j} < ℓ$ : conjugation transports $x_{r}^{m}$ by $s_{r} x_{r}^{m} s_{r} = x_{r + 1}^{m} - \sum_{k = 0}^{m - 1} x_{r}^{k} x_{r + 1}^{m - 1 - k} s_{r},$ so the leading term $x_{r}^{ℓ}$ produces $x_{r + 1}^{ℓ}$ plus correction terms of total degree $ℓ - 1$ , and every old remainder, when straightened, stays of degree below $ℓ$ .

3.1step 2.2algebra

Termination of the nondegenerate reduction. Order exponent vectors lexicographically as $(a_{d}, a_{d - 1}, \dots, a_{1})$ , larger first differing coordinate larger; this is a well-order on $N^{d}$ preserved by addition. Suppose $a_{r} \geq ℓ$ for some $r$ and some spanning monomial $X^{a} T_{w}$ . Multiplying the zero relation $R_{r}$ of step 2.2 on the left by $X^{a - ℓ e_{r}}$ and on the right by $T_{w}$ gives a relation whose leading term is $q_{H}^{r - 1} X^{a} T_{w}$ and whose remaining terms all have the same coordinates above $r$ and a strictly smaller $r$ -th coordinate (increases in coordinates below $r$ do not affect the lexicographic comparison), with $w \in S_{r}$ and finite straightening changing no dot exponents. Since $q_{H}^{r - 1}$ is a unit, this solves for $X^{a} T_{w}$ as a combination of spanning monomials with strictly smaller exponent vector; well-founded induction on the lexicographic order terminates and expresses every element using only monomials with every $a_{r} < ℓ$ . Hence the bounded ordered monomials span the nondegenerate quotient, over $A$ and after any base change.

3.2step 2.3algebra

Termination of the degenerate reduction. If $a_{r} \geq ℓ$ , multiply the zero relation $R_{r}$ of step 2.3 on the left by $x^{a - ℓ e_{r}}$ and on the right by $w$ ; the leading term is $x^{a} w$ , and every other term has total degree strictly below $\sum_{j} a_{j}$ (all other coefficients come from the remainder of $R_{r}$ , of degree $< ℓ$ ). Induction on the total degree $\sum_{j} a_{j}$ therefore expresses every element in terms of monomials $x^{a} w$ with all $a_{r} < ℓ$ . This also shows that no relation $f (x_{r}) = 0$ for $r > 1$ is claimed: at level one $f (t) = t$ , and $x_{1} = 0$ gives $x_{2} = s_{1}$ , which is exactly the form of the reduced spanning family.

4.1step 2.1step 3.1step 3.2∎

Conclusion. Steps 1.1, 2.1 and 3.1 give the spanning statement for the nondegenerate quotient by $f (X_{1}) = 0$ , and steps 1.1, 2.3 and 3.2 for the degenerate quotient by $f (x_{1}) = 0$ , with all exponents $a_{r} < ℓ$ in both branches and no independence asserted. Since the spanning families have at most $ℓ^{d} d!$ elements, each quotient has dimension at most $ℓ^{d} d!$ over a field. The specialisations $f (t) = \prod_{i} (t - q_{H}^{i})^{λ_{i}}$ and $f (t) = \prod_{i} (t - i)^{λ_{i}}$ are instances of the two branches with $ℓ = \sum_{i} λ_{i}$ . For $ℓ = 0$ the imposed relation is $1 = 0$ , giving the zero algebra for $d > 0$ and $A$ for $d = 0$ , as stated.

Remarks

The lemma is a pure spanning statement: no independence of the bounded ordered monomials is asserted here (the later cyclotomic basis theorem supplies it in the finite-dimensional setting). The two straightening identities are the only place where the difference between the nondegenerate and degenerate branches matters, and the reduction terminates by a well-order in the nondegenerate case and by total degree in the degenerate case. No choice principle is used.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Degenerate affine Hecke ordered monomials form a basis

Statement

Let $F$ be any field and let $H_{d}^{\deg}$ be the algebra generated by commuting $x_{1}, \dots, x_{d}$ and symmetric-group generators $s_{1}, \dots, s_{d - 1}$ with $s_{r} x_{r} = x_{r + 1} s_{r} - 1, s_{r} x_{r + 1} = x_{r} s_{r} + 1, s_{r} x_{j} = x_{j} s_{r} (j \notin {r, r + 1}),$ together with the symmetric-group quadratic, braid and distant commutation relations, as in the degenerate clause of Affine type-A presentations and charged cyclotomic quotients (The finite symmetric group $S_{n}$ , one-line notation, and cycle notation, The symmetric group has the Coxeter presentation). Then ${x_{1}^{a_{1}} \dots x_{d}^{a_{d}} w : a_{j} \geq 0, w \in S_{d}}$ is an $F$ -basis of $H_{d}^{\deg}$ . No cyclotomic basis or Brundan–Kleshchev isomorphism is assumed. For $d = 0$ the basis is the singleton empty monomial.

Facts & Assumptions

Given: A field $F$ , the polynomial ring $P = F [z_{1}, \dots, z_{d}]$ , its fraction field $K = F (z_{1}, \dots, z_{d})$ , and the crossed product $K ⋊ S_{d}$ with basis the group elements and multiplication $(f σ) (g τ) = f σ (g) σ τ$ .

[F1]

$H_{d}^{\deg}$ is presented by the relations of the Statement (Affine type-A presentations and charged cyclotomic quotients).

[F2]

$S_{d} = ⟨ s_{1}, \dots, s_{d - 1} ⟩$ with the Coxeter relations, and every word in the $s_{r}$ reduces to a standard word $w \in S_{d}$ ; a proper subword of a reduced expression has strictly smaller length (The symmetric group has the Coxeter presentation).

Proof

technique · direct

Given: The presentations of [F1], the field $F$ , the crossed product

$K ⋊ S_{d}$ , and the operator model of step 1.1.

1.1givenconstruct

The operator model. Let $S_{r}$ be the automorphism of $K$ swapping $z_{r}, z_{r + 1}$ and fixing the other variables, and put $δ_{r} : = z_{r} - z_{r + 1}$ . Let $z_{j}$ act on $K$ by multiplication and let $A_{r} : = S_{r} + \frac{S_{r} - 1}{δ_{r}} = (1 + \frac{1}{δ_{r}}) S_{r} - \frac{1}{δ_{r}} .$ For $p \in P$ the difference $S_{r} p - p$ is divisible by $δ_{r}$ in $P$ (write $p$ in the variables $z_{r}, z_{r + 1}$ with the remaining variables as scalars and use that $z_{r}^{a} z_{r + 1}^{b} - z_{r + 1}^{a} z_{r}^{b}$ is divisible by $δ_{r}$ ), so $A_{r}$ preserves $P$ ; hence the operators $z_{j}$ (multiplication) and $A_{r}$ generate a subalgebra of ${End}_{F} (K)$ and, restricted to $P$ , of ${End}_{F} (P)$ .

2.1F1algebra

Verification of the defining relations in the model. Direct substitution gives the mixed relations $A_{r} z_{r} - z_{r + 1} A_{r} = - 1$ , $A_{r} z_{r + 1} - z_{r} A_{r} = 1$ , and $A_{r} z_{j} = z_{j} A_{r}$ for $j \notin {r, r + 1}$ . Writing $a (t) : = 1 + 1 / t$ and $b (t) : = - 1 / t$ , so that $A_{r} = a (δ_{r}) S_{r} + b (δ_{r})$ , and using $S_{r} (δ_{r}) = - δ_{r}$ , the identity coefficient of $A_{r}^{2}$ is $a (δ_{r}) a (- δ_{r}) + b (δ_{r})^{2} = 1$ and its $S_{r}$ -coefficient is $a (δ_{r}) b (- δ_{r}) + b (δ_{r}) a (δ_{r}) = 0$ , so $A_{r}^{2} = 1$ ; distant $A_{r}$ commute because their permutations and coefficients involve disjoint variable pairs. For the braid relation put $u : = z_{r} - z_{r + 1}$ , $v : = z_{r + 1} - z_{r + 2}$ , $w : = u + v = z_{r} - z_{r + 2}$ , $S : = S_{r}$ , $T : = S_{r + 1}$ ; expanding $A_{r} A_{r + 1} A_{r}$ and $A_{r + 1} A_{r} A_{r + 1}$ in $K ⋊ S_{d}$ gives the six permutation coefficients $- \frac{u v + 1}{u v w} (1), \frac{a (u)}{v w} (S), \frac{a (v)}{u w} (T), - \frac{a (u) a (w)}{v} (S T), - \frac{a (v) a (w)}{u} (T S), a (u) a (w) a (v) (S T S),$ and the right expansion has the same six coefficients with $u, v$ and $S, T$ interchanged, so the two expansions agree. These are rational identities valid in every characteristic, including $2$ . Hence the assignment $z_{j} \mapsto$ multiplication by $z_{j}$ , $s_{r} \mapsto A_{r}$ defines an algebra homomorphism $H_{d}^{\deg} \to {End}_{F} (K)$ (and into ${End}_{F} (P)$ by step 1.1).

3.1F1F2step 2.1algebra

Spanning. In the model, $s_{r} p = (S_{r} p) s_{r} - (p - S_{r} p) δ_{r}^{- 1}$ for every polynomial $p$ , so every word in the generators rewrites as a linear combination of the ordered monomials $z^{a} w$ with $a_{j} \geq 0$ and $w \in S_{d}$ , the power-polynomial part being collected by commutativity and the $s$ -word reduced by [F2]. The same straightening performed in $H_{d}^{\deg}$ (available there because the model satisfies the defining relations, or by the same elementary computation in the presentation) shows that the ordered monomials span the algebra.

3.2F2step 2.1algebra

Independence. In the crossed-product expansion of a reduced product $A_{w}$ , the coefficient at $w$ is $c_{w} = \prod_{k = 1}^{m} (1 + 1 / (s_{i_{1}} \dots s_{i_{k - 1}}) (δ_{i_{k}})) \neq 0$ ; every other permutation has length less than $ℓ (w)$ , because it is represented by a proper subword. The distinct field automorphisms are independent over $K$ : from a shortest nonzero operator relation $\sum_{j} f_{j} σ_{j} = 0$ , compose with multiplication by a variable on which some $σ_{j}$ differs from $σ_{1}$ , and subtract the original relation multiplied by $σ_{1}$ of that variable. This gives a shorter nonzero relation; a one-term relation cannot annihilate $1$ . Thus the crossed product acts faithfully on $K$ . Now an $F$ -linear relation among the ordered monomials maps to $\sum_{w} f_{w} (z) A_{w} = 0$ , with $f_{w} (z) = \sum_{a} c_{a, w} z^{a} \in P$ . If some $f_{w}$ is nonzero, choose such a $w$ of maximal length. Its coefficient is $f_{w} c_{w} \neq 0$ and receives no contribution from any other maximal-length label or shorter label, a contradiction. Therefore every $f_{w} = 0$ , and polynomial coefficient uniqueness gives every $c_{a, w} = 0$ . This proves $F$ -linear independence of the ordered monomials, without asserting their impossible $K$ -linear independence.

4.1step 3.1step 3.2∎

Conclusion. Steps 3.1 and 2.1 show that the family ${x_{1}^{a_{1}} \dots x_{d}^{a_{d}} w : a_{j} \geq 0, w \in S_{d}}$ both spans and is linearly independent in $H_{d}^{\deg}$ over $F$ ; renaming the commuting generators $z_{j} \mapsto x_{j}$ and using the presentation [F1] gives the asserted $F$ -basis. No cyclotomic quotient, BK isomorphism or dimension theorem is used, and for $d = 0$ the family consists of the empty monomial alone.

Remarks

The model $A_{r} = S_{r} + (S_{r} - 1) / δ_{r}$ is the standard rational-operator model of the degenerate affine Hecke algebra; the only inversions are of the nonzero differences $δ_{r}$ , and the construction is valid in every characteristic. The independence proof uses finitely many automorphisms of a finitely generated field, so no choice principle is needed.

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Selfadjoint weight Gram valuations detect Jantzen classes

Statement

Let $O$ be a discrete valuation ring with uniformizer $π$ and residue field $F$ , and let $M$ be a finite free $O$ -module with an $O$ -valued bilinear form $b$ whose scalar extension to $Frac (O)$ is nondegenerate, as in DVR cellular Jantzen filtration and exact Gram valuation (the "(DVR leaf)"). Suppose $1 = \sum_{ν} e_{ν}$ is a finite decomposition into pairwise orthogonal idempotents of an $O$ -algebra $A$ acting on $M$ , with $e_{ν}^{*} = e_{ν}$ for an anti-involution $*$ satisfying $b (a x, y) = b (x, a^{*} y)$ . Write $M_{ν} : = e_{ν} M$ and let $G_{ν}$ be the restricted Gram matrix. Then each $M_{ν}$ is finite free, each restricted generic form is nondegenerate, and $\sum_{s > 0} \dim_{F} e_{ν} M_{F}^{s} = ν_{π} (\det G_{ν}) .$ For a split finite-dimensional highest-weight algebra $S_{F}$ whose standard classes $[Δ (μ)]$ form a $Z$ -basis of $K_{0}$ and whose matrix $K_{μ ν} = \dim e_{ν} Δ (μ)$ has a square unitriangular submatrix containing every standard row $μ$ and a selected set of partition-weight columns $ν$ of the same cardinality (Grothendieck group of an essentially small abelian category), the weight-character map $ch ⁣ : K_{0} \to Z^{{weights}}, [N] \mapsto (\dim e_{ν} N)_{ν},$ is injective. Consequently a class equality involving Jantzen pieces is equivalent to the corresponding equalities of weight Gram valuations.

Facts & Assumptions

Given: The DVR $O$ with uniformizer $π$ and residue field $F$ , the finite free $O$ -module $M$ with nondegenerate generic form $b$ , the orthogonal idempotent decomposition $1 = \sum_{ν} e_{ν}$ with $e_{ν}^{*} = e_{ν}$ , and the summands $M_{ν} = e_{ν} M$ .

[F1]

(DVR leaf) For the free module $M$ with form $b$ one has $\sum_{s > 0} \dim_{F} M_{F}^{s} = ν_{π} (\det G)$ for any Gram matrix $G$ , the reduced filtration $M_{F}^{s}$ terminates, and the subspace $M_{F}^{1}$ is the radical of the reduced form (DVR cellular Jantzen filtration and exact Gram valuation).

[F2]

Modules carry the unital actions of Unital left and right modules over a ring; unqualified module means left module. Submodule freeness and exactness of $N \mapsto e_{ν} N$ are proved locally below.

[F3]

By hypothesis the standard classes form a $Z$ -basis and the indicated finite weight submatrix contains every standard row and is unitriangular. The Grothendieck-group relation is additivity on short exact sequences (Grothendieck group of an essentially small abelian category); triangular invertibility is proved in step 1.4.

Proof

technique · direct

Given: The data of the Statement and [F1]–[F3].

1.1F2algebra

The decomposition is direct and its summands are free. The map $M \to ⨁_{ν} M_{ν}$ , $x \mapsto (e_{ν} x)_{ν}$ , is an $O$ -linear isomorphism with inverse $(x_{ν})_{ν} \mapsto \sum_{ν} x_{ν}$ (the $e_{ν}$ are orthogonal idempotents summing to $1$ ), so $M = ⨁_{ν} M_{ν}$ . To prove freeness, induct on $r$ for an arbitrary submodule $N \subseteq O^{r}$ . The assertion is immediate for $r = 0$ . The first-coordinate image is an ideal $J$ ; if nonzero, the minimum valuation among its nonzero elements is attained by some $a \in J$ , and every element of $J$ is divisible by $a$ , so $J = a O$ . Lift $a$ to $u \in N$ . Subtracting multiples of $u$ gives $N = O u \oplus \ker ({pr}_{1} ∣_{N})$ , with the kernel a submodule of $O^{r - 1}$ ; if $J = 0$ , all of $N$ lies in $O^{r - 1}$ . The induction proves finite freeness and rank at most $r$ . Apply it to each $M_{ν} \subseteq M$ .

1.2F2algebra

The summands are pairwise orthogonal and the restricted forms are nondegenerate. For $ν \neq ρ$ and $x, y \in M$ , $b (e_{ν} x, e_{ρ} y) = b (x, e_{ν}^{*} e_{ρ} y) = b (x, e_{ν} e_{ρ} y) = 0$ by $e_{ν}^{*} = e_{ν}$ and orthogonality, so $M_{ν}$ and $M_{ρ}$ are orthogonal for $b$ . If $x \in M_{ν}$ is orthogonal to $M_{ν}$ , then it is orthogonal to every $M_{ρ}$ (orthogonality) and hence to $M$ , so $x = 0$ by generic nondegeneracy of $b$ ; the identical orthogonality argument after scalar extension handles every generic vector, and thus the restriction of $b$ to $M_{ν}$ is nondegenerate over $Frac (O)$ .

1.3F2F3algebra

Exactness and the character map. The functor $N \mapsto e_{ν} N$ on $A$ -modules is exact: a morphism commutes with $e_{ν}$ , so its kernel on $e_{ν} N$ is $e_{ν}$ times its kernel on $N$ ; if $y = e_{ν} y$ lies in the image of a quotient map, any preimage $x$ may be replaced by $e_{ν} x$ with the same image. Hence $ch$ is an additive map on the Grothendieck group $K_{0}$ by [F3].

1.4F3algebra

Injectivity of the weight character. Write a class in $K_{0}$ as $\sum_{μ} a_{μ} [Δ (μ)]$ with $a_{μ} \in Z$ . Its coordinates on the partition weights are the row vector $a = (a_{μ})$ multiplied by the matrix $K = (K_{μ ν})$ , $K_{μ ν} = \dim e_{ν} Δ (μ)$ . Restricting only the columns to the selected partition weights retains every coefficient $a_{μ}$ ; the resulting square unitriangular matrix is invertible over $Z$ by finite triangular elimination (all diagonal entries are $1$ ); if $ch (a) = 0$ , then the coordinates on that submatrix vanish and the triangular system forces $a_{μ} = 0$ for every $μ$ . Hence $ch$ is injective.

2.1F1step 1.1step 1.2

The Jantzen filtration decomposes. For each $s$ the orthogonality of the summands gives $M (s) = ⨁_{ν} M_{ν} (s)$ with $M_{ν} (s) = {x \in M_{ν} : b (x, M_{ν}) \subseteq π^{s} O}$ , because the components of $x$ in the other summands pair trivially; consequently $M_{F}^{s} = ⨁_{ν} (M_{ν} (s) + π M_{ν}) / π M_{ν} = ⨁_{ν} e_{ν} M_{F}^{s}$ . For a zero-rank summand $M_{ν} = 0$ , each $e_{ν} M_{F}^{s} = 0$ , its empty Gram determinant is $1$ as specified by the DVR leaf, and both sides of the identity are $0$ . Applying the DVR leaf [F1] to the free module $M_{ν}$ with its restricted form and Gram matrix $G_{ν}$ gives the displayed equality $\sum_{s > 0} \dim_{F} e_{ν} M_{F}^{s} = ν_{π} (\det G_{ν})$ .

3.1step 2.1step 1.4∎

Conclusion. By step 2.1, the multiplicities appearing in a Jantzen class equality are exactly the weight Gram valuations $ν_{π} (\det G_{ν})$ ; by step 1.4 the weight character detects class equalities, so a class equality involving Jantzen pieces holds if and only if the corresponding equalities of weight Gram valuations hold. No Schur functor, residue-sequence character, rationalization or semisimple Hecke Grothendieck group is used anywhere.

Remarks

The lemma is the bridge that turns the exact DVR sum formula into a statement about characters: orthogonal selfadjoint idempotents split both the module and its Jantzen filtration, and a unitriangular standard-multiplicity matrix makes the weight character faithful. The argument is finite and choice-free.

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Symmetric polynomial centers have primitive split content idempotents

Statement

Let $A$ be a finite-dimensional unital algebra over a field $F$ (Algebras over a commutative ring, central structure maps, and algebra homomorphisms), with commuting elements $x_{1}, \dots, x_{d}$ and split joint regular-module generalized-eigenprojectors $p_{w o r d} \in A$ : the $p_{w o r d}$ are pairwise orthogonal, sum to $1$ , and $(x_{r} - {w o r d}_{r}) p_{w o r d}$ is nilpotent for every $r$ . For each finite multiset $α$ of eigenvalues let $e_{α} : = \sum_{w o r d of multiset α} p_{w o r d}$ , and assume each $e_{α}$ is central. If $Z (A)$ is the image of the symmetric polynomial ring in $x_{1}, \dots, x_{d}$ (Symmetric polynomials as the invariants of variable permutations, The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution), then each $e_{α}$ is zero or primitive central, and its nonzero terms are exactly all primitive central idempotents of $A$ (Blocks of a finite-dimensional algebra via primitive central idempotents). No splitting-simple-module hypothesis and no algebraic-closure hypothesis on $F$ is required. For $d = 0$ the asserted symmetric-center hypothesis says $Z (A) = F \cdot 1$ and yields the same conclusion.

Facts & Assumptions

Given: The finite commuting split-projector family, the central content idempotents, and the equality between $Z (A)$ and the symmetric-polynomial image, as assumed in the Statement.

[F1]

By hypothesis the joint generalized-eigenprojectors are orthogonal, sum to $1$ , and commute with every $x_{r}$ ; on each word summand the differences $x_{r} - {w o r d}_{r}$ are commuting nilpotents. The sums $e_{α}$ are central by hypothesis.

[F2]

A symmetric polynomial is invariant under all permutations of the variables (Symmetric polynomials as the invariants of variable permutations). Its elementary-symmetric generation is proved in step 1.2.

[F3]

A nonzero central idempotent is primitive if it cannot be split as a sum of two nonzero orthogonal central idempotents (Blocks of a finite-dimensional algebra via primitive central idempotents). The nilpotent-ideal criterion is proved in step 2.1.

Proof

technique · direct

Given: The data of the Statement and [F1]–[F3]. If $d = 0$ there is nothing

to prove beyond $Z (A) = F \cdot 1$ , whose only idempotents are $0$ and $1$ ; assume

$d \geq 1$ and fix a content $α$ with $e : = e_{α} \neq 0$ .

1.1F1algebra

The nilpotent ideal of the content summand. For $1 \leq k \leq d$ let $a_{k}$ be the value of $σ_{k}$ on any word of content $α$ (independent of the order of the word). On the summand $p_{w o r d} A$ inside $e A$ write $x_{r} = {w o r d}_{r} + N_{r}$ with commuting nilpotents $N_{r}$ by [F1]. Choose $M$ with $N_{r}^{M} = 0$ for all $r$ ; then every product of more than $d (M - 1)$ monomials of positive degree in the $N_{r}$ vanishes, because among more than $d (M - 1)$ factors some $N_{r}$ occurs at least $M$ times (finite pigeonhole). Hence every polynomial in the $N_{r}$ with zero constant term is nilpotent, and in particular $N_{k} : = e σ_{k} (x) - a_{k} e$ is nilpotent on every $p_{w o r d} A$ ; since there are finitely many words, a common exponent makes $N_{k}$ nilpotent in $e A$ . The $N_{k}$ commute, and the ideal $J$ they generate in the commutative ring $e Z (A)$ is nilpotent by the same degree count.

1.2F2algebra

Elementary-symmetric generation without division. For a homogeneous symmetric polynomial, order its finitely many monomials lexicographically with $x_{1} > \dots > x_{d}$ . Its leading exponent vector $(b_{1}, \dots, b_{d})$ has $b_{1} \geq \dots \geq b_{d}$ : otherwise swapping an inverted pair produces a larger monomial with the same coefficient, contrary to maximality. The product $σ_{1}^{b_{1} - b_{2}} σ_{2}^{b_{2} - b_{3}} \dots σ_{d}^{b_{d}}$ has leading monomial $x_{1}^{b_{1}} \dots x_{d}^{b_{d}}$ with coefficient $1$ . Subtract its multiple by the leading coefficient. The remaining polynomial is symmetric and has a smaller leading monomial of the same total degree; the finite set of degree- $m$ monomials makes repetition terminate. Apply this to each homogeneous component. Every symmetric polynomial is therefore a polynomial in the $σ_{k}$ , over any field and without division by factorials.

2.1F2F3step 1.1step 1.2algebra

The center of the content summand is a local ring. By step 1.2 every symmetric polynomial is a polynomial in $σ_{1}, \dots, σ_{d}$ , so the symmetric-center hypothesis gives $e Z (A) = F e + J .$ Since $e \neq 0$ and $J$ is nilpotent, $F e \cap J = 0$ : if $c e \in J$ with $c \neq 0$ , nilpotence would give $0 = (c e)^{M} = c^{M} e$ , contradicting $e \neq 0$ . Therefore $e Z (A) / J ≅ F$ , a field, and an idempotent $b$ has residue $0$ or $1$ in this field. In the first case $b \in J$ is nilpotent, and $b = b^{M} = 0$ ; in the second case the same argument applied to the idempotent $e - b \in J$ gives $b = e$ . Thus every idempotent of $e Z (A)$ is $0$ or $e$ .

3.1F1step 2.1∎

The nonzero content idempotents are exactly the primitive central idempotents. Step 2.1 shows that each $e_{α} \neq 0$ is a primitive idempotent of $Z (A)$ , hence primitive central in $A$ . Conversely, let $b$ be a primitive central idempotent of $A$ . The $e_{α}$ are central, pairwise orthogonal and sum to $1$ , so $b = \sum_{α} b e_{α}$ ; if two terms $b e_{α}$ were nonzero, they would be orthogonal nonzero idempotents summing to the primitive idempotent $b$ , a contradiction, so exactly one term equals $b$ . This nonzero $b e_{α}$ is an idempotent of $e_{α} Z (A)$ , hence equals $e_{α}$ by step 2.1, proving $b = e_{α}$ . Hence the nonzero $e_{α}$ are precisely the primitive central idempotents, and the blocks of $A$ are exactly the nonzero content summands. No algebraic closure, splitting-simple-module theorem, power-sum generation, division by $d!$ , or level-one restriction is used.

Remarks

The proof is the local center argument: the content idempotents make the center of each summand a local ring with nilpotent nilradical, so its idempotents can only be $0$ and the unit. Everything is finite and choice-free.

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Polynomial spectra and central content sectors of finite cyclotomic Hecke algebras

Statement

Use the charged finite- $e$ field specialization of Affine type-A presentations and charged cyclotomic quotients over any field $F$ . Then $H_{d}^{Λ}$ is finite-dimensional before any basis independence argument. Every polynomial generator has a monic annihilator split over the residue values $c_{i}$ ( $c_{i} = q_{H}^{i}$ in the nondegenerate branch, $c_{i} = i \cdot 1_{F}$ in the degenerate branch (Bounded ordered monomials span the cyclotomic Hecke quotients)). The canonical finite CRT projectors $ε_{r} (i)$ and $e (ν) = \prod_{r} ε_{r} (ν_{r})$ are orthogonal, sum to $1$ , and project every finite $H$ -module onto its joint generalized residue spaces. For each $α = \sum_{i} n_{i} α_{i}$ of height $d$ , $e_{α} = \sum_{ν of content α} e (ν)$ is central, and $H$ is the direct sum of its central ideals $e_{α} H$ , possibly zero; no primitivity is asserted here. On $e (ν)$ , $z_{r} - c_{ν_{r}}$ is nilpotent, so all invertible-constant local power series have finite evaluations.

Facts & Assumptions

Given: The charged finite- $e$ specialization over $F$ with a dominant weight $Λ$ of level $ℓ = \sum_{i} λ_{i}$ , the cyclotomic polynomial $f (t) = \prod_{i} (t - c_{i})^{λ_{i}}$ imposed on the first polynomial generator, and the rank- $d$ quotient $H_{d}^{Λ}$ .

[F1]

The presentations and relations of Affine type-A presentations and charged cyclotomic quotients hold; the nondegenerate parameter $q_{H}$ has multiplicative order $e$ , and in the degenerate branch $q_{H} = 1$ and $char F = e$ with distinct parameters $c_{i} = i \cdot 1_{F}$ (Bounded ordered monomials span the cyclotomic Hecke quotients).

[F2]

The cyclotomic quotients are spanned by bounded ordered monomials, with all exponents $0 \leq a_{r} < ℓ$ , in both branches (Bounded ordered monomials span the cyclotomic Hecke quotients).

[F3]

The adjugate identity $adj (M) M = \det (M) I = M adj (M)$ holds over every commutative ring, and its determinant has the leading coefficient computed from the entries (For every positive-sized square matrix over a commutative ring, $A adj (A) = adj (A) A = \det (A) I$ , Division algorithm for polynomials over a field).

[F4]

For pairwise comaximal ideals the Chinese remainder theorem gives orthogonal idempotents summing to $1$ and an isomorphism onto the product (Chinese remainder theorem for pairwise comaximal ideals).

Proof

technique · direct

Given: The charged specialization, its cyclotomic quotient $H = H_{d}^{Λ}$ ,

and the data of [F1]–[F4]; write $q = q_{H}$ for the nondegenerate parameter.

1.1F1algebra

Normal spanning in the affine algebra. In the degenerate branch the mixed relation $s_{r} p = (s_{r} p) s_{r} - (p - s_{r} p) (x_{r} - x_{r + 1})^{- 1}$ moves every polynomial past a simple reflection at the cost of its permuted term and a polynomial lower term; in the nondegenerate branch the relations $X_{r} T_{r} = T_{r} X_{r + 1} + (1 - q) X_{r + 1}$ and $X_{r + 1} T_{r} = T_{r} X_{r} + (q - 1) X_{r + 1}$ , together with their inverse-variable forms, move every Laurent variable to the left. Each correction lowers the number of misplaced variable/crossing pairs, so the process terminates; the $T$ -words reduce to the finitely many standard words $T_{w}$ by the Hecke relations, and in the degenerate branch the $s$ -words reduce to the permutations $w$ . Hence the affine algebra is spanned by the ordered monomials $X^{a} T_{w}$ ( $a \in Z^{d}$ , $w \in S_{d}$ ) in the nondegenerate branch and by $x^{a} w$ ( $a \in Z_{\geq 0}^{d}$ ) in the degenerate branch. No independence and no injection of the finite Hecke algebra is assumed.

1.2F1algebra

Global intertwiners. For neighboring polynomial variables $x, y$ the degenerate element $θ : = s (x - y) + 1$ satisfies $θ x = y θ$ and $θ^{2} = 1 - (x - y)^{2}$ ; the nondegenerate element $Θ : = T (1 - x y^{- 1}) + 1 - q$ satisfies $Θ x = y Θ$ and $Θ^{2} = (x - q y) (y - q x) / (x y)$ . Both are identities in the whole affine algebra: substituting the mixed relations and the quadratic relation (using $T^{- 1} = q^{- 1} T + q^{- 1} - 1$ in the nondegenerate case) and clearing the nonzero factors gives exactly the displayed products. Hence they remain valid in the cyclotomic quotient. These are the source's global identities for the two branches, proved here by substitution rather than imported.

2.1F3step 1.2algebra

Monic propagation. Suppose the split monic polynomial $f$ of degree $D$ annihilates the old variable $x$ . Multiplying the corresponding identity of step 1.2 by $f (x)$ and using $f (x) = 0$ gives $f (y) θ^{2} = 0 or f (y) Θ^{2} = 0,$ i.e. $f (y) r (x, y) = 0$ with $r$ the respective quadratic in $x, y$ (in the nondegenerate case after multiplying by the commuting unit $x y$ ). Let $C : = F [z] / (f (z))$ with power basis $1, z, \dots, z^{D - 1}$ and consider multiplication by $r (z, y)$ on $C [y]$ . Its matrix $M (y)$ has leading coefficient $- Id$ in the degenerate case and $- q Id$ in the nondegenerate case; hence $\det M (y)$ has invertible leading coefficient. The adjugate identity [F3], applied to the basis vector $1$ , gives $r (z, y) adj (M (y)) = \det M (y)$ ; lifting this identity from $C [y]$ back to $F [z, y]$ and using the defining relation of $C$ , then multiplying by $f (y)$ and using $f (y) r (x, y) = 0$ , yields $f (y) \det M (y) = 0$ . Dividing by the invertible leading scalar produces a monic annihilator of the next variable. No flatness, algebraic closure, generic faithfulness or denominator difference is used.

3.1step 2.1algebra

Spectral inclusion. If $f (z) = \prod_{a} (z - a)^{m_{a}}$ is split with roots $a$ , the same power-basis determinant factors as $\prod_{a} r (a, y)$ with the multiplicities $m_{a}$ : the multiplication matrix is triangular after decomposing $C$ into the finite factors $(z - a)^{m_{a}}$ (or by the companion-matrix resultant identity). In the degenerate branch the new roots are $a + 1, a - 1$ , and in the nondegenerate branch they are $q a, q^{- 1} a$ . Starting from $\prod_{i} (x_{1} - i)^{λ_{i}}$ therefore produces only roots in the prime-field residue set ${0, 1, \dots, e - 1}$ (closed under $a \mapsto a \pm 1$ ), and starting from $\prod_{i} (X_{1} - q^{i})^{λ_{i}}$ produces only roots in ${q^{i} : i \in I}$ (closed under $a \mapsto q^{\pm 1} a$ ). Multiplicities are retained and no square-free assumption is made; in the nondegenerate branch all roots are nonzero, so every annihilator has nonzero constant coefficient and expresses the inverse variable as a polynomial in that variable.

3.2F2step 1.1step 2.1algebra

Finiteness. Applying step 2.1 successively along the positions shows that every polynomial generator has a finite-degree monic annihilator; its monic relation bounds the exponents of that generator, and in the nondegenerate branch it also eliminates negative exponents. The bounded normal spanning of [F2], together with the normal spanning of step 1.1, therefore expresses every element of the quotient through finitely many exponent tuples and the finitely many $T_{w}$ (respectively $w$ ). Hence $H$ is finite-dimensional over $F$ , without using the exact $ℓ^{d} d!$ basis theorem or any independence of the bounded monomials.

4.1F3F4step 3.2

Canonical CRT projectors. The minimal polynomial of each polynomial generator on its regular module divides the monic split annihilator constructed in steps 2.1 and 3.1, so it is split with only the required residues $c_{i}$ as roots. For each position $r$ the pairwise comaximal ideals $((z_{r} - c_{i})^{m_{r, i}})_{i}$ give, by the finite polynomial division algorithm and the Chinese remainder theorem [F4], canonical orthogonal projectors $ε_{r} (i)$ in the finite-dimensional algebra generated by $z_{r}$ ; these project every finite module onto its generalized $c_{i}$ -eigenspace, including the case of a zero factor. Their commuting products $e (ν) = \prod_{r} ε_{r} (ν_{r})$ are orthogonal and sum to $1$ , and act as the simultaneous generalized residue projectors. On the summand $e (ν)$ the element $z_{r} - c_{ν_{r}}$ is nilpotent by construction, so every power series with invertible constant term has a finite evaluation there.

5.1F1step 1.2step 4.1algebra

Centrality of the content sectors. Using the Hecke cross and intertwiner identities of step 1.2, the only possible weights of the image of an element supported on a joint generalized residue $ν$ under a neighboring Hecke generator are $ν$ and $s_{r} ν$ ; a fixed-content sum $e_{α} = \sum_{ν of content α} e (ν)$ is therefore invariant under every generator, hence central. The central idempotents $e_{α}$ are orthogonal and sum to $1$ , so $H$ is the direct sum of its central ideals $e_{α} H$ , some of which may be zero. Primitivity of the $e_{α}$ is not asserted or inferred from this argument.

6.1step 3.2step 4.1step 5.1∎

Conclusion. Steps 1.1 and 3.2 give finite-dimensionality before any basis independence; steps 2.1 and 3.1 give the monic split annihilators with the stated residue values; step 4.1 gives the canonical orthogonal CRT projectors and the nilpotence of $z_{r} - c_{ν_{r}}$ on $e (ν)$ ; and step 5.1 gives centrality of the content sectors $e_{α}$ with $H = ⨁_{α} e_{α} H$ . No primitivity, blockwise BK isomorphism, cyclotomic exact-dimension theorem or global-basis claim is used.

Remarks

The lemma isolates the finite/split-spectral input of the residue construction: monic annihilators propagate along the strands by a power-basis determinant argument, which requires only the adjugate identity over a commutative ring, and the CRT then produces the canonical residue projectors. The centrality argument uses the global intertwiners rather than any primitivity of the content sectors. No choice principle is used.

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Central content sectors of the cyclotomic Hecke algebra

Definition

Work in the charged finite- $e$ specialization of Affine type-A presentations and charged cyclotomic quotients at a dominant weight $Λ$ and rank $d$ , and let $H = H_{d}^{Λ}$ . For the canonical CRT projectors supplied by Polynomial spectra and central content sectors of finite cyclotomic Hecke algebras, denote by $e (ν)$ the joint residue projector attached to the residue tuple $ν$ and by $e_{α} : = \sum_{ν of content α} e (ν)$ their sum at content $α$ (Affine type-A presentations and charged cyclotomic quotients).

Content sector. Put $H_{α}^{Λ} : = e_{α} H_{d}^{Λ} = e_{α} H_{d}^{Λ} e_{α},$ an ideal and unital algebra with unit $e_{α}$ , called the central content sector of content $α$ ; it may be the zero algebra, and at this stage it is NOT asserted to be a primitive block. The two descriptions coincide because $e_{α}$ is a central idempotent.

Named nilpotent generators. On $H_{α}^{Λ}$ put $y_{r} : = \sum_{ν of content α} (x_{r} - ν_{r}) e (ν) (degenerate branch), y_{r} : = \sum_{ν of content α} (1 - q_{H}^{- ν_{r}} X_{r}) e (ν) (nondegenerate branch) .$ By the spectral supplier these named elements are commuting nilpotents on the sector, and they provide the local variables in which the sector is compared with the corresponding KLR block.

Boundary conventions. For $α = 0$ one has $e_{e m p t y} = 1$ and $H_{0}^{Λ} = F$ at rank zero. Later primitivity statements may identify the nonzero sectors with the blocks of $H$ , but that identification does not change the objects named here. This definition introduces no new primitivity or basis assertion of its own and uses no choice principle.

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Charged core height matrices: hub-preserving descent and minimal switch connectivity

Statement

Fix finite $e \geq 2$ , $ℓ \geq 1$ and integer charges $c_{1}, \dots, c_{ℓ}$ . Give the nodes of component $a$ the residue $c_{a} + j - i (m o d e)$ . Write $C_{i}$ for the residue counts, $δ_{i}$ for removable minus addable $i$ -nodes, and $m_{i} : = # {a : c_{a} \equiv i}$ . Then $δ_{i} = 2 C_{i} - C_{i - 1} - C_{i + 1} - m_{i},$ with both neighbouring terms retained (and counted twice) at $e = 2$ . Equal hubs and equal size are together equivalent to equal content. For componentwise $e$ -cores let $b_{i a}$ be the largest charged beta bead on runner $i$ and $t_{i a} : = \frac{b_{i a} - i}{e} .$ Cores of fixed hub and charges correspond exactly to integer matrices with fixed row sums and column sums $c_{a} - e$ ; for two such cores the size difference is $\frac{e}{2}$ times the difference of $\sum t_{i a}^{2}$ . Every nonminimum-size core admits a finite same-hub core-switch path that never increases size and ends at strictly smaller size, and every pair of minimum-size cores is connected by equal-size switches. A switch subtracts $1$ at $(i, a), (j, b)$ and adds $1$ at $(i, b), (j, a)$ ; each path switch can be oriented with $γ = t_{i a} - t_{i b} - t_{j a} + t_{j b} \geq 2$ , giving size change $- e (γ - 2)$ . Minimum cores exist for every nonempty fixed-hub class.

Facts & Assumptions

Given: Finite $e \geq 2$ , $ℓ \geq 1$ , integer charges $c_{a}$ , the charged beta conventions of The e-core, the e-quotient, and the e-weight, and the column-potential lemma Ultra-tight multisets admit integer column potentials.

[F1]

For a partition $λ$ with charge $c$ , the beta set $B = {λ_{i} + c - i : i \geq 1}$ contains all sufficiently negative integers and no sufficiently positive integer, and has charge $c$ : $# (B \cap [- L, \infty)) - L = c$ for all sufficiently large integers $L$ . Conversely a set with these tail conditions and charge $c$ determines a partition by ordering its elements $β_{1} > β_{2} > \dots$ and putting $λ_{i} = β_{i} - c + i$ . Indeed the tail and charge imply $β_{i} = c - i$ eventually, while $β_{i} - β_{i + 1} \geq 1$ makes the resulting $λ_{i}$ weakly decreasing, eventually zero and therefore nonnegative (Partitions, English diagrams, and conjugation, The e-core, the e-quotient, and the e-weight).

[F2]

Removing rim hooks and the componentwise $e$ -core are independent of the removal order, and the $e$ -weight is an invariant (The e-core is independent of the rim hook removal order, The e-core, the e-quotient, and the e-weight).

[F3]

For a finite multiset in $L_{ℓ} = Z^{ℓ} / Z (1, \dots, 1)$ tightness, zero exchanges and ultra-tightness are as in Ultra-tight multisets admit integer column potentials, and ultra-tight multisets are exactly those admitting integer column potentials (Ultra-tight multisets admit integer column potentials).

Proof

technique · direct

Given: The charges $c_{a}$ , the components and their beta sets, and the data

of [F1]–[F3].

1.1F1F2algebra

Beta sets, bead moves and the rim-hook dictionary. By [F1] a bead move $u \mapsto u - h$ with $u \in B$ , $u - h \notin B$ unwraps a rim hook of length $h$ whose foot has residue $u - h + 1$ modulo $e$ : if $β_{i} = u$ and $β_{j} < u - h < β_{j - 1}$ , the rows $i, \dots, j - 1$ change by the boundary between those two beta positions, the lost border strip has $h$ boxes, and its lowest-row leftmost lost column makes the foot content $u - h + 1$ ; reversing the move wraps a hook with foot residue the starting beta position plus one. Moving a bead one position upward on its runner therefore removes an $e$ -hook, and packing every runner upward leaves the number of beads relative to the negative reference tail on each runner unchanged; by [F2] the result is the componentwise $e$ -core independent of the order. Every $e$ -hook contains one node of each residue, so hook removal preserves the hub (the residue-count vector modulo the all-one vector) and subtracts the all-one vector from the residue counts.

1.2F1algebra

The boundary-count identity. Let $C_{i}$ be the residue counts and $δ_{i}$ the number of removable minus addable $i$ -nodes. We prove $δ_{i} = 2 C_{i} - C_{i - 1} - C_{i + 1} - m_{i}$ by induction while adjoining nodes to the empty multipartition. The empty multipartition has $C_{i} = 0$ for all $i$ and $δ_{i} = - m_{i}$ (each component contributes one addable first node of its residue $c_{a}$ in the empty diagram), which gives the identity initially. Adjoining a node of residue $j$ changes the boundary count by $+ 2$ at $j$ and $- 1$ at each of $j - 1$ and $j + 1$ (counted twice when $e = 2$ ); these changes are read off by recording the occupied and vacant beta positions before and after the corresponding one-position bead move, all other adjacent occupancies cancelling. The right-hand side changes in exactly the same way because $C_{j}$ increases by $1$ and $C_{i}$ for $i \neq j$ is unchanged. Induction on the number of nodes gives the identity for every multipartition.

1.3F1F2algebra

Cores and their height matrices. For a componentwise $e$ -core let $b_{i a}$ be the largest bead on runner $i$ ( $0 \leq i < e$ ) and $t_{i a} = (b_{i a} - i) / e$ , so the packed beta set on runner $i$ is ${i + e k : k \leq t_{i a}}$ . Counting beads $\geq - e L$ for large $L$ gives the column sums $\sum_{i} t_{i a} = c_{a} - e$ . Writing $R_{i} = \sum_{a} t_{i a}$ , the hub satisfies $δ_{i} = R_{i} - R_{i - 1}$ for $1 \leq i < e$ and $δ_{0} = R_{0} - R_{e - 1} - ℓ$ ; since $\sum_{i} R_{i} = \sum_{a} c_{a} - e ℓ$ is fixed, equal hub is equivalent to equal row sums. Hence cores of a fixed hub and fixed charges are exactly the integer $e \times ℓ$ matrices $T = (t_{i a})$ with fixed row sums $R_{i}$ and column sums $c_{a} - e$ .

1.4F3algebra

Switches and energy. Let $T$ be an integer $e \times ℓ$ matrix with fixed margins, and consider a switch in distinct rows $i, j$ and distinct columns $a, b$ , replacing $(t_{i a}, t_{i b}, t_{j a}, t_{j b})$ by $(t_{i a} - 1, t_{i b} + 1, t_{j a} + 1, t_{j b} - 1)$ ; margins are preserved. With $γ : = t_{i a} - t_{i b} - t_{j a} + t_{j b}$ , the squared-norm changes by $4 - 2 γ$ : so $γ = 2$ is a zero-energy switch and $γ \geq 3$ strictly lowers the energy. Regard the $e$ labelled rows of $T$ as a multiset in $L_{ℓ} = Z^{ℓ} / Z (1, \dots, 1)$ ; it is tight exactly when $γ \leq 2$ for all choices, and switches with $γ = 2$ are exactly the zero exchanges of [F3].

2.1step 1.2algebra

Equal hub and equal size give equal content. If two multipartitions have equal hubs, the difference $d_{i}$ of their residue counts satisfies $2 d_{i} - d_{i - 1} - d_{i + 1} = 0$ by step 1.2, because the $δ_{i}$ and $m_{i}$ terms are unchanged; hence all first differences $d_{i} - d_{i - 1}$ are equal, and summing them around the cycle makes this common difference $0$ , so $d$ is constant. Therefore equal hub and equal size imply equal content, and equal hubs imply sizes congruent modulo $e$ .

2.2F1step 1.3algebra

The size formula. If $T (κ), T (η)$ are the height matrices of two such cores, then $∣ η ∣ - ∣ κ ∣ = \frac{e}{2} (\sum_{i, a} t_{i a} (η)^{2} - \sum_{i, a} t_{i a} (κ)^{2}) .$ Indeed, count the beta numbers $\geq - e L$ in a component: their number is $e L + c_{a}$ , the same for both cores, while their sum changes by $\sum_{i} i Δ t_{i a} + \frac{e}{2} \sum_{i} Δ (t_{i a}^{2} + t_{i a})$ ; from $β_{j} = λ_{j} + c_{a} - j$ the change in that sum equals the change in the partition size, and summing over $a$ the equal row sums kill $\sum_{i} i \sum_{a} Δ t_{i a}$ while the equal column sums kill the linear $\sum Δ t_{i a}$ term. In particular the squared-norm difference is even, so the displayed size difference is an integer multiple of $e$ .

3.1F3step 1.3step 2.2step 1.4algebra

Ultra-tight rows and the global minimum. If the row multiset is not ultra-tight, a finite sequence of zero switches leads to a non-tight matrix, at which some rows and columns have $γ \geq 3$ , giving a strict descent: every intermediate matrix has the same margins, zero switches preserve core size by step 2.2, and a strict descent lowers core size by $e (γ - 2) \geq e$ . If instead the multiset is ultra-tight, [F3] supplies a column potential $s_{a}$ with every row of $U = (t_{i a} - s_{a})$ having maximum minus minimum at most one. For a fixed row sum such a row minimizes the sum of squares over all integer rows: if two entries differ by at least $2$ , transferring one from the larger to the smaller lowers the square sum by at least $2$ , and repeated transfers terminate; the minimum is attained exactly by rows whose entries differ by at most one. Column potentials do not affect comparisons of total squared norm between matrices with the same column sums, because $\sum_{i, a} (t_{i a} - s_{a})^{2} = \sum_{i, a} t_{i a}^{2} - 2 \sum_{a} s_{a} \sum_{i} t_{i a} + e \sum_{a} s_{a}^{2} .$ Hence such a $T$ is a global minimum among the matrices of its margins, and conversely a global minimum admits no strict descent after zero switches, so its row multiset is ultra-tight. Starting from any $T$ , repeated zero-path and strict-descent steps reach a global minimum, since all matrices represent genuine cores of nonnegative integer size and strict descent lowers that size by at least $e$ ; minima therefore exist for every nonempty fixed-hub class.

4.1F3step 3.1algebra

Equal-size connectivity of minima. Let $T, V$ be two global minima with the same margins, and let $s$ be the column potential of $T$ from step 3.1. Every row of $U = T - s$ is balanced, with entries $u_{i}$ or $u_{i} + 1$ ; since $V - s$ has the same row sums and the same total energy, each of its rows also attains the row minimum and has entries $u_{i}$ or $u_{i} + 1$ . Subtracting $u_{i}$ from row $i$ produces two binary $e \times ℓ$ matrices $A, B$ with identical row and column sums. If $A \neq B$ , the difference $D = A - B$ has entries $0, \pm 1$ and zero row and column sums; starting from a $+ 1$ and alternating a $- 1$ in the same row with a $+ 1$ in the same column produces a shortest alternating cycle with distinct rows $i_{0}, \dots, i_{t - 1}$ and columns $j_{0}, \dots, j_{t - 1}$ , where $D_{i_{k} j_{k}} = + 1$ and $D_{i_{k} j_{k + 1}} = - 1$ . If $t = 2$ , switch the rectangle in $A$ to agree with $B$ there. If $t > 2$ , inspect $A_{i_{0} j_{t - 1}}$ and $B_{i_{0} j_{t - 1}}$ : if $A_{i_{0} j_{t - 1}} = 0$ , switching the rectangle with rows $i_{0}, i_{t - 1}$ and columns $j_{0}, j_{t - 1}$ gives a binary $A^{'}$ with strictly fewer discrepancies from $B$ ; if $B_{i_{0} j_{t - 1}} = 1$ , the reverse switch on $B$ gives a binary $B^{'}$ with strictly fewer discrepancies from $A$ ; if neither holds, $D_{i_{0} j_{t - 1}} = + 1$ and a shorter alternating cycle exists, contradicting minimality. Induction on the discrepancy count connects $A$ and $B$ by binary switches, all preserving margins. Lifting these binary switches back to $T$ gives switches with $γ = 2$ , hence equal core size by step 2.2, preserving the hub.

5.1step 3.1step 2.1step 4.1∎

Conclusion. Step 1.2 proves the boundary-count identity $δ_{i} = 2 C_{i} - C_{i - 1} - C_{i + 1} - m_{i}$ with the double edge at $e = 2$ , and step 2.1 shows that equal hub and equal size are equivalent to equal content. Steps 1.3 and 2.2 identify cores of fixed hub and charges with integer matrices of fixed margins and give the squared-norm size formula; step 3.1 provides a same-hub core-switch path that never increases size and strictly decreases it whenever the core is not of minimum size, with minima existing in every nonempty class; and step 4.1 connects any two minimum-size cores by equal-size switches. These are exactly the finite- $e$ descent and reduced-core connectivity facts consumed by the Lyle–Mathas max-weight argument, proved locally without importing weight nonnegativity or matrix-alignment inputs.

Remarks

The energy argument is the normalized replacement for Fayers' weight-theory proof: the column potential makes the minimal rows balanced, every switch is scored by $γ - 2$ , and zero exchanges never change the core size, so the combinatorics of minimal cores reduces to the ultra-tight multiset lemma and the binary-switch connectivity argument. All steps are finite. No choice principle is used.

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Cyclic type-A KLR regular Hom norm gives the total dimension

Statement

Let $F$ be any field, $I = Z / e Z$ with finite $e \geq 2$ , and use the standard homogeneous cyclotomic KLR quotient at $Λ = \sum_{i} λ_{i} Λ_{i}$ for the cyclic affine type-A Cartan datum and admissible graded parameters (The cyclic affine type A Cartan datum and its residues, The cyclotomic KLR quotient at a dominant integral weight). Put $ℓ = \sum_{i} λ_{i}$ and $B_{d} = ⨁_{ht β = d} R^{Λ} (β) .$ Then $\dim_{F} B_{d} = ℓ^{d} d!$ for $d > 0$ , and $B_{0} = F$ has dimension one. The proof uses the basic first induction-restriction adjunction and the mixed relations on projective $K_{0}$ modulo $v - 1$ , where $v$ is the formal grading variable, independent of any Hecke parameter $q_{H}$ . No Frobenius theorem, tensor-form perfectness, highest-weight or global-basis isometry, or Hecke basis is required. If $ℓ = 0$ and $d > 0$ , then $B_{d} = 0$ .

Open obligation. The proof consumes the sibling suppliers Cyclotomic induction is projective on the right (step 3.1), The categorical ${s l}_{2}$ commutator for cyclotomic KLR functors (step 2.1) and Normalized cyclotomic functors give an integrable quantum action (step 2.1); if any of these is not yet authored, this item's decision stays escalated until the supplier and the exact use are reconciled.

Facts & Assumptions

Given: The cyclic affine type-A Cartan datum with finite $e \geq 2$ , the dominant weight $Λ$ of level $ℓ$ , the standard cyclotomic KLR quotient $R^{Λ} = ⨁_{β} R^{Λ} (β)$ , its rank pieces $B_{d}$ , and the induction and restriction functors $F_{i}^{b a s i c}, E_{i}^{b a s i c}$ (right append and its adjoint) (Cyclotomic KLR induction and restriction functors, The cyclotomic KLR quotient at a dominant integral weight).

[F1]

The corner idempotents $e (β, i)$ are orthogonal and sum to the local unit $1_{B_{d}}$ ; left multiplication by $R^{Λ} (β + i) e (β, i)$ gives an isomorphism $B_{d} e (β, i) \otimes_{R^{Λ} (β)} R^{Λ} (β) ≅ B_{d} e (β, i)$ with inverse $b \mapsto b \otimes 1$ , using only the unital corner homomorphism $A \to e A e$ (Cyclotomic KLR induction and restriction functors, The cyclotomic KLR quotient at a dominant integral weight).

[F2]

On projective $K_{0}$ modulo $v - 1$ the same-color categorical decomposition forgets to $[E_{i}^{b a s i c}, F_{i}^{b a s i c}] = h_{i} (Λ - β) Id$ on the root component $β$ , and the unequal-color mixed relation forgets to $[E_{i}^{b a s i c}, F_{j}^{b a s i c}] = 0$ for $i \neq j$ (The categorical ${s l}_{2}$ commutator for cyclotomic KLR functors, Normalized cyclotomic functors give an integrable quantum action).

[F3]

The rank- $d$ basic projectives are $B_{d}$ itself, all restricted modules remain finite projective, and the first adjunction $F_{i}^{b a s i c} ⊣ E_{i}^{b a s i c}$ holds (Cyclotomic induction is projective on the right, Graded Grothendieck groups, shift action, and Cartan map).

[F4]

For finite graded projectives the pairing $h (P, Q) = \dim_{F} Hom (P, Q)$ with grading forgotten (equivalently the total graded Hom Hilbert series evaluated at $v = 1$ ) is additive, invariant under shifts and descends to the $K_{0}$ quotient of [F3] (Graded Grothendieck groups, shift action, and Cartan map).

Proof

technique · direct

Given: The cyclic KLR data of [F1]–[F4], the operators

$D = \sum_{i} F_{i}^{b a s i c}$ , $E = \sum_{i} E_{i}^{b a s i c}$ on projective

$K_{0}$ modulo $v - 1$ , and the classes $P_{d} = [B_{d}]$ , $P_{0} = [F]$ .

1.1F1F3algebra

Rank decomposition. For fixed $d$ , the corner idempotents $e (β, i)$ over all root components $β$ of height $d - 1$ and colors $i$ are orthogonal and sum to $1_{B_{d}}$ : each word has exactly one last color and one prefix root, so it lies in exactly one corner. By [F1] the multiplication map $⨁_{β, i} B_{d} e (β, i) \otimes_{R^{Λ} (β)} R^{Λ} (β) \to B_{d}$ is an isomorphism; spelling out the left action by the basic induction functor gives $P_{d} = D P_{d - 1} = D^{d} P_{0}$ . The zero-corner case gives $B_{d} e (β, i) = 0$ and is included since its summand is zero.

1.2F2algebra

The mixed commutators at $v = 1$ . Forgetting the grading shifts in the same-color categorical decomposition gives $[E_{i}^{b a s i c}, F_{i}^{b a s i c}] = h_{i} (Λ - β) Id$ on the root component $β$ ; its positive identity summands contribute $h_{i}$ copies and its negative identity summands contribute the signed counterpart, so the integral $[h_{i}]_{v}$ specializes at $v = 1$ to the integer $h_{i}$ with no division by $v - v^{- 1}$ . The unequal-color mixed relation gives $[E_{i}^{b a s i c}, F_{j}^{b a s i c}] = 0$ for $i \neq j$ . Hence $[E, D] = \sum_{i} [E_{i}^{b a s i c}, F_{i}^{b a s i c}] = (\sum_{i} h_{i} (Λ - β)) Id$ on root component $β$ . For affine type A every Cartan column sums to zero, including the double-edge matrix at $e = 2$ , so $\sum_{i} h_{i} (Λ - β) = \sum_{i} λ_{i} - \sum_{j} β_{j} \sum_{i} a_{i j} = ℓ$ independently of $β$ . Therefore $[E, D] = ℓ \cdot Id$ , and since rank-zero restriction is zero, $E P_{0} = 0$ ; expanding $[E, D^{d}] = d ℓ D^{d - 1}$ gives $E P_{d} = d ℓ P_{d - 1}$ .

2.1F3F4step 1.1step 1.2algebra

The Hom-norm recurrence. For finite graded projectives define $h$ as in [F4]; it is additive in both variables, unchanged by shifts, and descends to projective $K_{0}$ modulo $v - 1$ . Pairing distinct root blocks by zero and summing over a fixed rank, the ordinary first adjunction $F_{i}^{b a s i c} ⊣ E_{i}^{b a s i c}$ gives $h (F_{i}^{b a s i c} P, Q) = h (P, E_{i}^{b a s i c} Q)$ , hence $h (D P, Q) = h (P, E Q)$ ; all restricted modules remain finite projectives by [F3], so the recurrence stays inside the declared group. Combining with step 1.2, $h (P_{d}, P_{d}) = h (D P_{d - 1}, P_{d}) = h (P_{d - 1}, E P_{d}) = d ℓ h (P_{d - 1}, P_{d - 1}) .$

3.1step 1.1step 2.1∎

Conclusion. The initial value is $h (P_{0}, P_{0}) = \dim_{F} {End}_{F} (F) = 1$ ; since ${End}_{B_{d}} (B_{d}) ≅ B_{d}^{o p}$ by right multiplication, $h (P_{d}, P_{d}) = \dim_{F} B_{d}$ , and induction on $d$ gives $\dim_{F} B_{d} = ℓ^{d} d!$ as an ordinary integer. Nothing reduces the integer dimension or the factor $d ℓ$ inside $F$ , so the argument covers every field characteristic; for $ℓ = 0$ and $d > 0$ the recurrence gives $\dim_{F} B_{d} = 0$ ; and for $d = 0$ , independently of $ℓ$ , $B_{0} = F$ has the singleton empty monomial as basis. No Frobenius theorem, tensor-form perfectness, highest-weight or global-basis isometry, Hecke basis or choice principle is used.

Remarks

This is the independent dimension input used by the later exact basis theorem: the rank pieces satisfy the recurrence $h (P_{d}, P_{d}) = d ℓ h (P_{d - 1}, P_{d - 1})$ inside projective $K_{0}$ , so the total dimension $ℓ^{d} d!$ is obtained without any global basis or Hecke-theoretic input. The three sibling suppliers named in the Statement remain an open obligation until they are authored and their uses reconciled.

PropositionStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Cyclotomic Hecke algebras of level one are finite Hecke algebras

Statement

Let $Λ = Λ_{j}$ be a level-one fundamental weight and keep the conventions of Affine type-A presentations and charged cyclotomic quotients. Then the assignments $T_{r} \mapsto T_{r}$ ( $1 \leq r < d$ ) in the nondegenerate case and $s_{r} \mapsto s_{r}$ in the degenerate case extend to $F$ -algebra isomorphisms $H_{F, q} (S_{d}) ⟶ H_{d}^{Λ}, F S_{d} ⟶ H_{d}^{Λ}$ respectively (The generic type-A Hecke algebra, The standard basis of the generic type-A Hecke algebra).

Facts & Assumptions

Given: A field $F$ , the level-one fundamental weight $Λ = Λ_{j}$ , the cyclotomic quotient $H_{d}^{Λ}$ of Affine type-A presentations and charged cyclotomic quotients, and the finite type-A Hecke algebra $H_{F, q} (S_{d})$ with quadratic relation $T_{r}^{2} = (q - 1) T_{r} + q$ , $q = q_{H}$ , respectively the group algebra $F S_{d}$ .

[F1]

At level one the cyclotomic relation reads $X_{1} = q^{j}$ in the nondegenerate branch and $x_{1} = j \cdot 1_{F}$ in the degenerate branch; the mixed relations give $X_{r + 1} = q^{- 1} T_{r} X_{r} T_{r}$ and $x_{r + 1} = s_{r} x_{r} s_{r} + s_{r}$ , so every polynomial generator is expressible in the first one and the finite generators (Affine type-A presentations and charged cyclotomic quotients).

[F2]

The finite Hecke algebra has standard basis ${T_{w} : w \in S_{d}}$ ; the Jucys–Murphy elements $L_{r} = \sum_{a < r} (a r)$ in $F S_{d}$ commute and satisfy $s_{r} L_{r + 1} = L_{r} s_{r} + 1$ and $s_{r} L_{t} = L_{t} s_{r}$ for $t \notin {r, r + 1}$ , with the analogous multiplicative identities in $H_{F, q} (S_{d})$ (The Hecke Jucys-Murphy action is dominance triangular, The standard basis of the generic type-A Hecke algebra).

Proof

technique · direct

Given: The level-one quotient $H_{d}^{Λ}$ and the finite generators

$T_{r}$ (respectively $s_{r}$ ).

1.1F1algebra

Surjectivity of the natural map. By [F1] the first polynomial generator is the scalar $q^{j}$ (respectively $j$ ), and the recursion expressing every $X_{r + 1}$ (respectively $x_{r + 1}$ ) in $X_{r}$ and $T_{r}$ (respectively $s_{r}$ ) shows that the algebra generated by the finite generators and $X_{1}$ is already generated by the finite generators alone. Hence the assignment of the Statement is a surjective algebra homomorphism.

1.2F1F2construct

The inverse in the nondegenerate branch. Let $L_{r}$ be the additive Jucys–Murphy element and put $M_{r} : = 1 + (q - 1) L_{r} = q^{1 - r} T_{r - 1} \dots T_{1}^{2} \dots T_{r - 1},$ a product of units, so every $M_{r}$ is invertible; the $M_{r}$ commute because the $L_{r}$ do, and $T_{r} M_{r} T_{r} = q M_{r + 1}$ while $T_{r} M_{t} = M_{t} T_{r}$ for $t \notin {r, r + 1}$ follow from [F2] by separating the term $a = r$ in $L_{r + 1}$ and using the finite braid relation on the three affected transpositions. Setting $X_{r} : = q^{j} M_{r}$ therefore produces commuting units with $T_{r} X_{r} T_{r} = q X_{r + 1}$ , distant mixed commutation, the finite Hecke relations and $X_{1} = q^{j}$ . These elements satisfy the complete affine presentation, so the assignment defines a homomorphism $H_{d}^{Λ} \to H_{F, q} (S_{d})$ fixing the $T_{r}$ .

1.3F1F2construct

The inverse in the degenerate branch. Put $x_{r} : = j + L_{r}$ with $L_{r} = \sum_{a < r} (a r) \in F S_{d}$ . Direct transposition multiplication gives $s_{r} L_{r + 1} s_{r} = L_{r} + s_{r}$ , hence $s_{r} x_{r + 1} = x_{r} s_{r} + 1$ , and the $x_{r}$ commute and commute with distant generators by [F2]; at $r = 1$ this gives $x_{1} = j$ . These elements satisfy the degenerate affine relations, so the assignment defines a homomorphism $H_{d}^{Λ} \to F S_{d}$ fixing the $s_{r}$ .

2.1step 1.1step 1.2step 1.3algebra∎

Mutual inverses. The two constructions of steps 1.2 and 1.3 fix every finite generator, and the composites with the surjection of step 1.1 fix every presentation generator: the finite generators by construction, and the polynomial generators because both sides satisfy the same recursion from $X_{1} = q^{j}$ (respectively $x_{1} = j$ ) and the mixed relations. Hence both composites are the identity on a generating set, so the maps are mutually inverse isomorphisms; rank zero gives $F$ on both sides. This removes any late-basis or dimension dependency and introduces no BK or primitive-block input.

Remarks

The identification is proved by an explicit inverse rather than by comparing dimensions, so it does not depend on the later cyclotomic basis theorem, the BK isomorphism or any block-primitivity statement. No choice principle is used.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Affine and cyclotomic type-A Hecke algebras

Definition

Let $F$ be a field and $q \in F^{\times}$ with finite quantum characteristic $e = e (F, q)$ , $2 \leq e < \infty$ (The quantum characteristic of an invertible Hecke parameter), let $I = Z / e Z$ be the residue set with integer residue labels $0, 1, \dots, e - 1$ , and use the cyclic affine type-A Cartan datum with dominant weights $P^{+}$ and pairing $(Λ, α_{i})$ (The cyclic affine type A Cartan datum and its residues). In the nondegenerate branch the distinct spectral values of the labels are $q^{i}$ , while in the degenerate branch they are $i \cdot 1_{F}$ (here $e = char F$ ). Fix $Λ \in P^{+}$ of level $ℓ = \sum_{i \in I} (Λ, α_{i})$ and put $λ_{i} : = (Λ, α_{i}) \geq 0$ .

(NG) Nondegenerate case $q \neq 1$ . The affine Hecke algebra of type $A_{d - 1}$ at parameter $q$ is the unital $F$ -algebra $H_{d}$ with generators $X_{1}^{\pm 1}, \dots, X_{d}^{\pm 1}$ and $T_{1}, \dots, T_{d - 1}$ subject to the relations $X_{r} X_{s} = X_{s} X_{r}, X_{r} X_{r}^{- 1} = 1,$ $T_{r} X_{r} T_{r} = q X_{r + 1}, T_{r} X_{s} = X_{s} T_{r} (s \notin {r, r + 1}),$ $T_{r}^{2} = (q - 1) T_{r} + q, T_{r} T_{r + 1} T_{r} = T_{r + 1} T_{r} T_{r + 1}, T_{r} T_{s} = T_{s} T_{r} (∣ r - s ∣ > 1) .$ The cyclotomic Hecke algebra is $H_{d}^{Λ} : = H_{d} / ⟨ \prod_{i \in I} (X_{1} - q^{i})^{λ_{i}} ⟩ .$

(DG) Degenerate case $q = 1$ (which forces $char F = e > 0$ ). The degenerate affine Hecke algebra $H_{d}$ is the unital $F$ -algebra with generators $x_{1}, \dots, x_{d}$ and $s_{1}, \dots, s_{d - 1}$ subject to the relations $x_{r} x_{s} = x_{s} x_{r}, s_{r} x_{r + 1} = x_{r} s_{r} + 1, s_{r} x_{s} = x_{s} s_{r} (s \notin {r, r + 1}),$ $s_{r}^{2} = 1, s_{r} s_{r + 1} s_{r} = s_{r + 1} s_{r} s_{r + 1}, s_{r} s_{t} = s_{t} s_{r} (∣ r - t ∣ > 1);$ its ordered monomials $x_{1}^{m_{1}} \dots x_{d}^{m_{d}} w$ form an $F$ -basis by Degenerate affine Hecke ordered monomials form a basis. The degenerate cyclotomic Hecke algebra is $H_{d}^{Λ} : = H_{d} / ⟨ \prod_{i \in I} (x_{1} - i)^{λ_{i}} ⟩ .$

Conventions and boundary. Put $H_{0}^{Λ} = F$ ; the cyclotomic polynomial is imposed on the FIRST polynomial generator, and $T_{r}$ acts on positions $r, r + 1$ . The Hecke parameter is the one of The generic type-A Hecke algebra ( $T_{i}^{2} = (q - 1) T_{i} + q$ ). At level one $Λ = Λ_{j}$ , the algebra $H_{d}^{Λ}$ is the finite type-A Hecke algebra $H_{F, q} (S_{d})$ if $q \neq 1$ and the group algebra $F S_{d}$ if $q = 1$ (Cyclotomic Hecke algebras of level one are finite Hecke algebras). Only finite $e$ is treated: the source's $e = 0$ branches ( $q$ not a root of unity, and the characteristic-zero degeneration, whose quiver is $A_{\infty}$ ) are outside this page's cyclic scope and are recorded as a deferred boundary in the batch notes. These presentations and the charged quotients are the ones of Affine type-A presentations and charged cyclotomic quotients; no embedding of the label set $I$ into $F$ is claimed in the nondegenerate branch, and no choice principle is used.

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The type-A cyclotomic specialization of the QG-KLR algebras

Definition

Let $F, q, e, I, Λ, λ_{i}$ be as in Affine and cyclotomic type-A Hecke algebras (The quantum characteristic of an invertible Hecke parameter, The cyclic affine type A Cartan datum and its residues). On the cyclic affine type-A Cartan datum take the normalized symmetrizers $d_{i} = 1$ and define the type-A KLR datum $Q^{A}$ by $Q_{i i}^{A} = 0, Q_{i j}^{A} = 1 if a_{i j} = 0,$ $Q_{i, i + 1}^{A} (u, v) = v - u, Q_{i + 1, i}^{A} (u, v) = u - v (e \geq 3),$ $Q_{01}^{A} (u, v) = Q_{10}^{A} (u, v) = - (u - v)^{2} (e = 2) .$ Write $R_{α}$ for the resulting generalized KLR algebra over $F$ (The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum, The KLR grading and the diagrammatic multiplication) and $R_{α}^{Λ}$ for the standard cyclotomic quotient of The cyclotomic KLR quotient at a dominant integral weight (monic first-strand relations $x_{1}^{λ_{i_{1}}} e (ν) = 0$ ).

The specialization dictionary. Put $y_{r} : = x_{r}$ and $ψ_{r} : = - τ_{r}$ . With these normalizations, the relations (1.6)–(1.15) of the source hold in $R_{α}^{Λ}$ , and conversely the quotient presented by those relations is the same algebra; in other words $R_{α}^{Λ}$ is the sign-modified cyclotomic Khovanov–Lauda algebra of the source for the cyclic quiver of quantum characteristic $e$ . The verification of the dictionary is elementary and is recorded here so that no hidden convention is imported:

(Q1)–(Q4). Non-adjacent pairs have $a_{i j} = 0$ and $Q_{i j}^{A} = 1$ , homogeneous of degree $0$ . For an adjacent pair $(i, i + 1)$ with $e \geq 3$ one has $b_{i j} = a_{i j} = - 1$ , and $v - u$ is homogeneous of degree $2 = - 2 b_{i j}$ with nonzero extremal coefficients of $u^{- a_{i j}} = u$ and $v^{- a_{j i}} = v$ ; the opposite orientation is the transpose, so (Q2) is built into the two formulae. For $e = 2$ the pair ${0, 1}$ has $a_{01} = a_{10} = - 2$ and $b_{01} = - 2$ , and $- (u - v)^{2}$ is symmetric, homogeneous of degree $4 = - 2 b_{01}$ , with top coefficient $- 1 \neq 0$ .
Dot sliding. Relation (1.8) comes from $τ_{r} e (i) = e (s_{r} i) τ_{r}$ ; relations (1.12)–(1.13) reduce to the dot-sliding constants $τ_{a} x_{a} - x_{a + 1} τ_{a} = e (ν)$ and $τ_{a} x_{a + 1} - x_{a} τ_{a} = - e (ν)$ on equal colours, whose sign changes to the source's constants $+ 1$ under $ψ = - τ$ .
Quadratic and braid relations. Relation (1.14): equal colours give $τ^{2} = Q_{i i} = 0$ ; non-adjacent colours give $τ^{2} = 1 = e (i)$ ; a forward arrow gives $τ^{2} = v - u$ , evaluated at $(y_{r}, y_{r + 1})$ as $y_{r + 1} - y_{r}$ ; a backward arrow gives $y_{r} - y_{r + 1}$ ; the $e = 2$ double edge gives $(y_{r + 1} - y_{r}) (y_{r} - y_{r + 1})$ . Relation (1.15): the divided-difference numerator $[Q^{A} (x_{a}, x_{a + 1}) - Q^{A} (x_{a + 2}, x_{a + 1})] / (x_{a + 2} - x_{a})$ equals $\pm 1$ in the two single-edge cases with $i_{r + 2} = i_{r}$ , equals $y_{r} + y_{r + 2} - 2 y_{r + 1}$ in the $e = 2$ double-edge case, and vanishes otherwise; the global sign $ψ = - τ$ matches the source's three cases. The product order is explicit: the source difference is $ψ_{r} ψ_{r + 1} ψ_{r} - ψ_{r + 1} ψ_{r} ψ_{r + 1}$ , which under $ψ = - τ$ equals the native reversed-word difference $τ_{r + 1} τ_{r} τ_{r + 1} - τ_{r} τ_{r + 1} τ_{r}$ ; for $Q = - (u - v)^{2}$ the displayed quotient is $u + w - 2 v$ . Thus no braid-order sign is suppressed, also in characteristic two.

Degrees of products are read on the successive permuted colour pairs as in The KLR grading and the diagrammatic multiplication. This definition introduces no new basis, trace or primitive-block assertion, and uses no choice principle.

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Cyclotomic Hecke intertwiners and local power series

Statement

Let $H_{α}^{Λ}$ be a central content sector (possibly zero) as in Central content sectors of the cyclotomic Hecke algebra, with the residue tuples $i = (i_{1}, \dots, i_{d})$ contained in a fixed index set $I_{α}$ .

Set $y_{r} : = \sum_{i \in I_{α}} (x_{r} - i_{r}) e (i)$ in the degenerate case and $y_{r} : = \sum_{i \in I_{α}} (1 - q^{- i_{r}} X_{r}) e (i)$ in the nondegenerate case ( $1 \leq r \leq d$ ). Then $y_{1}, \dots, y_{d}$ are commuting nilpotent elements of $H_{α}^{Λ}$ ; consequently every rational expression displayed below, expanded as a power series in the nilpotent $y$ 's, is a finite and well-defined element of $H_{α}^{Λ}$ .
The power series $p_{r} (i) : = {\begin{cases} 1, & i_{r} = i_{r + 1}, \\ (i_{r} - i_{r + 1} + y_{r} - y_{r + 1})^{- 1}, & i_{r} \neq i_{r + 1} \end{cases}$ (degenerate case), respectively $P_{r} (i) : = {\begin{cases} 1, & i_{r} = i_{r + 1}, \\ (1 - q) (1 - y_{r} (i) y_{r + 1} (i)^{- 1})^{- 1}, & i_{r} \neq i_{r + 1}, \end{cases} y_{r} (i) : = q^{i_{r}} (1 - y_{r})$ (nondegenerate case), satisfy (3.23)–(3.25) respectively (4.24)–(4.26), (4.28)–(4.30) and (4.32); and the intertwining elements $φ_{r} : = \sum_{i} (s_{r} + p_{r} (i)) e (i), Φ_{r} : = \sum_{i} (T_{r} + P_{r} (i)) e (i)$ satisfy all identities (3.14)–(3.20) of the source's Lemma 3.1, respectively (4.14)–(4.20) of its Lemma 4.1.
There exist invertible power series $q_{r} (i) \in F [⁣ [y_{r}, y_{r + 1}] ⁣]$ (respectively $Q_{r} (i)$ ) satisfying (3.27)–(3.29), respectively (4.33)–(4.35); the explicit candidates (3.30) respectively (4.36) do, and for them the identity (3.43) respectively (4.44) holds.

Facts & Assumptions

Given: The cyclotomic Hecke algebra $H^{Λ} = H_{d}^{Λ}$ of Affine and cyclotomic type-A Hecke algebras with its central content sectors $H_{α}^{Λ}$ and projectors $e (i)$ (Central content sectors of the cyclotomic Hecke algebra), and the notation of the Statement.

[F1]

On the summand $e (i)$ the polynomial generator has the single generalized residue $i_{r}$ : in the degenerate branch $x_{r} - i_{r}$ is nilpotent on $e (i)$ , and in the nondegenerate branch $q^{- i_{r}} X_{r} - 1$ is nilpotent on $e (i)$ ; the $y_{r}$ are commuting sums of these, hence commuting nilpotents (Central content sectors of the cyclotomic Hecke algebra).

[F2]

The divided-difference convention is $\partial_{r} (f) = (s_{r} f - f) / (y_{r} - y_{r + 1})$ , and on equal colours $i_{r} = i_{r + 1}$ the intertwiners reduce to $φ_{r} = s_{r} + 1$ respectively $Φ_{r} = T_{r} + 1$ (Affine and cyclotomic type-A Hecke algebras).

Proof

technique · direct

Given: The data of the Statement and the facts above.

1.1F1algebra

Finite evaluation. On each projector $e (i)$ the dots are commuting nilpotents by [F1]. If $y_{r}^{n_{r}} = 0$ , every monomial of total degree greater than $\sum_{r} (n_{r} - 1)$ vanishes. Thus formal series have finite evaluation, and $c + f$ with $c \neq 0$ and $f$ of positive degree has inverse $c^{- 1} \sum_{k \geq 0} (- f / c)^{k}$ . Unequal-colour spectral differences have nonzero scalar constant; equal-colour dot differences will only be used as formal divided differences, never inverted in the cyclotomic algebra.

1.2F2algebra

A faithful formal calculation convention. Temporarily take independent dots $u_{1}, \dots, u_{d}$ and a separate copy of their fraction field for each colour word. Let $S_{r}$ swap both positions and their dots; write multiplication on the right of $S_{r}$ . For spectral variables $A, B$ , put $\hat{p} = (A - B)^{- 1}$ in DG and $\hat{P} = (1 - q) B / (B - A)$ in NG, even on equal colours. The affine Hecke operator is $H_{r} = S_{r} (1 - \hat{p}) - \hat{p}$ in DG and $H_{r} = S_{r} (1 - \hat{P}) - \hat{P}$ in NG. For precision the formal calculation algebra has coefficients $⨁_{i} F [⁣ [u_{1}, \dots, u_{d}] ⁣] e (i)$ and the affine generators, with spectral word-sliding: an unequal-colour intertwiner moves $e (i)$ to $e (s_{r} i)$ and permutes coefficients, while on equal colours the affine mixed relation gives its divided-difference correction. These relations follow in a sector by the CRT projectors and the mixed presentation: multiply the unequal spectral difference by the intertwiner and use its unit inverse; on equal colours use $H + 1$ . This calculation is faithful before any cyclotomic quotient: move coefficients right by these mixed and word-sliding relations and reduce crossing words by the Coxeter relations to span the finitely many $H_{w} e (i)$ ; in their crossed-product expansions the coefficient at a reduced permutation $w$ is a product of nonzero rational functions and every proper subword has smaller length. A maximal-length coefficient therefore proves independence, also for formal series coefficients after clearing their finitely many difference denominators. Consequently an identity between regular formal expressions can be checked in this rational model and then evaluated at nilpotent dots. This is a calculation in the affine presentation, not an assumption that its cyclotomic quotient acts faithfully.

2.1step 1.1algebra

Explicit normalization. Write $u = y_{r}, v = y_{r + 1}$ , $p = (i_{r} - i_{r + 1} + u - v)^{- 1}$ on unequal DG colours, and $A = q^{i_{r}} (1 - u), B = q^{i_{r + 1}} (1 - v)$ on NG colours. In the order equal, unrelated, forward arrow, backward arrow, double edge, choose $q_{r} (i)$ to be $1 + v - u$ , $1 - p$ , $(1 - p^{2}) / (v - u)$ , $1$ , $(1 - p) / (v - u)$ , respectively, and choose $Q_{r} (i)$ to be $1 - q + q v - u$ , $(A - q B) / (A - B)$ , $(A - q B) / (A - B)^{2}$ , $q^{i_{r}}$ , $q^{i_{r}} / (A - B)$ , respectively. These are the actual five branches, including $e = 2$ . For DG forward colours $i_{r} - i_{r + 1} = - 1$ , so $1 - p^{2} = (v - u) (2 - u + v) / (1 - u + v)^{2}$ ; on the double edge the field has characteristic two and $1 - p = (v - u) / (1 + u - v)$ . Thus the forward quotient has constant $2 \neq 0$ unless the pair is a double edge, which uses the separate constant $1$ branch. Unrelated pairs have $i_{r} - i_{r + 1} \neq 0, \pm 1$ , so $1 - p$ has nonzero constant. In NG the unequal $A - B$ is a unit; $A - q B$ has nonzero constant except for a backward arrow, whose coefficient is prescribed separately. Hence all five coefficients are units; the equal coefficient has constant $1$ in DG and $1 - q \neq 0$ in NG.

2.2step 1.2algebra

The full affine coefficient check. In left-coefficient notation $H_{A B} = a_{A B} S + b_{A B}$ , use $(h, t) = (1, 0)$ in DG and $(q, q - 1)$ in NG, where $b_{A B} = - (A - B)^{- 1}$ or $(1 - q) B / (A - B)$ , respectively, and $a_{A B} = h - b_{A B}$ . Direct common-denominator multiplication gives $b_{A B} + b_{B A} = t$ , $a_{A B} a_{B A} = h + t b_{A B} - b_{A B}^{2}$ , and $b_{A B} b_{B C} = b_{A C} (b_{A B} + b_{B C} - t)$ . The mixed relations follow by multiplying $A, B$ on either side; the first two identities give $H^{2} = t H + h$ . For three positions, expansion of either braid word has coefficients, in the order $1, S_{1}, S_{2}, S_{1} S_{2}, S_{2} S_{1}, S_{1} S_{2} S_{1}$ , equal to $b_{A C} (h + b_{A B} b_{B C})$ , $a_{A B} b_{B C} b_{A C}$ , $a_{B C} b_{A B} b_{A C}$ , $a_{A B} a_{A C} b_{B C}$ , $a_{B C} a_{A C} b_{A B}$ , $a_{A B} a_{A C} a_{B C}$ . For example the first expansion's constant is $b_{A B}^{2} b_{B C} + a_{A B} a_{B A} b_{A C}$ , which reduces to the displayed constant using the three scalar identities. The $S_{1}$ coefficient reduces from $a_{A B} (b_{A C} b_{B A} + b_{A B} b_{B C})$ in the same way. Thus this six-coefficient check establishes braid for all spectral triples, including any coincidence of colour labels; spectral variables remain independent at this stage. Disjoint positions commute.

3.1step 2.1algebra

Pair products and interchange. Denote the chosen normalizer by $N_{i j} (u, v)$ , the piecewise scalar by $p$ in DG or $P = (1 - q) B / (B - A)$ in NG, and put $K_{i j} (u, v) = 1, v - u, u - v, (v - u) (u - v)$ in the four unequal-colour cases. Substitution gives $p_{i j} + p_{j i} (v, u) = 0$ in DG and $P_{i j} + P_{j i} (v, u) = 1 - q$ in NG. Also $N_{i j} (u, v) N_{j i} (v, u) K_{i j} (u, v) = 1 - p_{i j}^{2}$ in DG and $(1 - P_{i j}) (q + P_{i j})$ in NG. For the latter equality, its right side is $(B - q A) (A - q B) / ((B - A) (A - B))$ ; forward and backward pairs use $B - q A = q^{i_{r + 1}} (u - v)$ and $A - q B = q^{i_{r}} (v - u)$ , respectively, and a double edge uses both. These substitutions prove each branch of the pair-product identities without division by a nonunit. Each coefficient depends only on its ordered two colours and their two dots. Therefore swapping a third position gives the same coefficient of the same ordered pair, proving the three-position interchange identities for $p, P, N$ . In particular no square root is chosen.

4.1step 3.1step 2.2algebra

The intertwiners. On unequal colours $φ = S (1 - p)$ and $Φ = S (1 - P)$ , since the scalar term cancels. Hence they permute the word and the two spectral variables, their squares are the pair products in step 3.1, and they commute at distant positions. On equal colours they are $H + 1$ ; their squares are $2 φ$ and $(1 + q) Φ$ . The equal-colour mixed correction is $1 + B - A$ in DG and $q B - A$ in NG, obtained by multiplying the operator $H + 1$ by $B$ and subtracting $A (H + 1)$ . The braid defect is zero unless the outer colours coincide. For all equal colours, expanding $(H_{1} + 1) (H_{2} + 1) (H_{1} + 1)$ minus its reversed word gives $h (φ_{1} - φ_{2})$ in DG and $q (Φ_{1} - Φ_{2})$ in NG (here $h = 1$ in DG). For outer equal, middle unequal, write $C_{A B} = 1 - {\hat{p}}_{A B}$ or $1 - {\hat{P}}_{A B}$ , and $F_{A B} = C_{A B} C_{B A}$ . The middle equal-colour intertwiner contributes either its permutation term or its scalar term $C_{A C}$ . The former gives equal longest-permutation coefficients in the two words; the latter gives identity coefficients $F_{A B} C_{A C}$ and $F_{B C} C_{A C}$ . Their difference is therefore $C_{A C} (F_{A B} - F_{B C})$ . In DG put $p_{1} = 1 / (A - B), p_{2} = 1 / (B - C)$ and use $p_{1} + p_{2} = (A - C) p_{1} p_{2}$ ; then this difference is $(p_{1} - p_{2}) (p_{1} p_{2} - p_{1} - p_{2})$ . In NG, substituting $C_{A C} = (A - q C) / (A - C)$ and $F_{A B} = (A - q B) (B - q A) / (- (A - B)^{2})$ cancels the factor $A - C$ and gives $(1 - q)^{2} (A C - B^{2}) (A B - q B C) / ((A - B)^{2} (B - C)^{2})$ . Adjacent-equal and distinct-colour patterns have no remaining identity term; their permutation coefficients cancel by the scalar identities of step 2.2. These are precisely the intertwiner identities named in the Statement.

5.1step 3.1step 4.1algebra

A local crossing calculation that will also serve both relation maps. Put $D_{r} f = (S_{r} f - f) / (u_{r} - u_{r + 1})$ ; every inverse dot difference in an operator formula acts by multiplication on the left unless explicitly included in a right coefficient. On equal colours, $H + 1 = (S + 1) (- N / (u - v))$ , since $1 - \hat{p} = - N / (u - v)$ in DG and $1 - \hat{P} = - N / (u - v)$ in NG; thus $(H + 1) N^{- 1} = D$ . On unequal colours set $c_{i j} (u, v) = (1 - p) / N_{i j}$ or $(1 - P) / N_{i j}$ , so the normalized crossing is $S_{r} c_{i j}$ and $c_{i j} (u, v) c_{j i} (v, u) = K_{i j} (u, v)$ by step 3.1. For a formal coefficient $f$ , direct subtraction gives $D f = (S f) D + (S f - f) / (u - v)$ and $D^{2} = 0$ ; unequal crossings simply permute coefficients. These formulas prove the dot-sliding and quadratic relations and the compatibility $ψ_{r}^{2} N_{r} (i) S_{r} N_{r} (s_{r} i) = 1 - p_{r} (i)^{2}$ or $(1 - P_{r} (i)) (q + P_{r} (i))$ on unequal colours; on equal colours both sides of the original compatibility identity are zero.

6.1step 5.1algebra

Every braid colour pattern. Write $u, v, w$ for the three dots, $W = S_{1} S_{2} S_{1}$ , and $c = c_{i k}$ . If all colours are distinct, both crossing braid words have right coefficient $c_{i j} (u, v) c_{i k} (u, w) c_{j k} (v, w)$ at $W$ . For $(i, i, k)$ both are $(W - S_{1} S_{2}) c (u, w) c (v, w) / (v - w)$ ; the coefficient product is symmetric in $u, v$ , so moving it through $D_{1}$ adds no divided-difference correction. For $(i, k, k)$ both are $(W - S_{2} S_{1}) c (u, v) c (u, w) / (u - v)$ , by the same calculation with the product symmetric in $v, w$ . In these two formulas the denominator multiplies on the left and the coefficient product multiplies on the right. For $(i, j, i)$ the longest-permutation terms cancel by the ordered-pair interchange of step 3.1; the remaining identity coefficients in $ψ_{1} ψ_{2} ψ_{1}$ and $ψ_{2} ψ_{1} ψ_{2}$ are $- K_{i j} (u, v) / (u - w)$ and $- K_{i j} (w, v) / (u - w)$ . Thus their difference is $(K_{i j} (w, v) - K_{i j} (u, v)) / (u - w)$ , equal to $0, 1, - 1, u + w - 2 v$ for unrelated, forward, backward, double-edge pairs. Finally all equal colours give $D_{1} D_{2} D_{1} = D_{2} D_{1} D_{2} = - (1 - S_{1} - S_{2} + S_{1} S_{2} + S_{2} S_{1} - W) / ((u - v) (u - w) (v - w))$ . Each displayed quotient is a formal divided difference; the resulting braid correction is polynomial, including characteristic two. This proves all five cases locally.

7.1step 1.1step 3.1step 4.1step 6.1∎

Regular identities and conclusion. The rational calculations above express every intertwiner, normalization, dot-sliding, quadratic and braid identity as an identity of regular formal expressions in the affine generators. Faithfulness in step 1.2 licenses clearing the temporary differences, and step 1.1 licenses finite evaluation in the sector. The candidate normalizers and their products give all stated identities, retaining $y_{r} = 1 - q^{- i_{r}} y_{r} (i)$ in NG. Therefore all three assertions of the Statement hold over the original field, with no algebraic closure or choice principle.

Remarks

The temporary difference inverses occur only in the faithful formal calculation. They are not evaluated in a cyclotomic quotient; only the regular identities and the unit normalizers are evaluated there.

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The cyclotomic KLR block satisfies the Hecke relations

Statement

In the specialized cyclotomic KLR content algebra $R_{α}^{Λ}$ define $x_{r} : = \sum_{i \in I_{α}} (y_{r} + i_{r}) e (i), s_{r} : = \sum_{i \in I_{α}} (ψ_{r} q_{r} (i) - p_{r} (i)) e (i)$ in the degenerate case and $X_{r} : = \sum_{i \in I_{α}} q^{i_{r}} (1 - y_{r}) e (i), T_{r} : = \sum_{i \in I_{α}} (ψ_{r} Q_{r} (i) - P_{r} (i)) e (i)$ in the nondegenerate case, where $p, P, q, Q$ are the power series of Cyclotomic Hecke intertwiners and local power series evaluated in $R_{α}^{Λ}$ using the nilpotence of the $y$ 's. Then these elements satisfy the defining relations (3.2)–(3.5), respectively (4.2)–(4.5), of the affine Hecke algebra and the cyclotomic relation $\prod_{i} (x_{1} - i)^{λ_{i}} = 0$ , respectively $\prod_{i} (X_{1} - q^{i})^{λ_{i}} = 0$ ; hence they define a unique $F$ -algebra homomorphism $σ ⁣ : H_{d}^{Λ} \to R_{α}^{Λ}$ . Moreover $σ$ kills $e_{β}$ and every corner component $e_{β} H_{d}^{Λ} e_{γ}$ with $(β, γ) \neq (α, α)$ , and restricts to a unital homomorphism $σ ⁣ : H_{α}^{Λ} \to R_{α}^{Λ}$ with $σ (e (i)) = e (i)$ for $i \in I_{α}$ and $σ (e (i)) = 0$ otherwise.

Facts & Assumptions

Given: The specialized cyclotomic KLR algebra $R_{α}^{Λ}$ with its idempotents $e (i)$ , dots $y_{r}$ and crossings $ψ_{r}$ (The type-A cyclotomic specialization of the QG-KLR algebras), the sector $H_{α}^{Λ}$ (Central content sectors of the cyclotomic Hecke algebra), and the power series $p_{r} (i), q_{r} (i)$ (degenerate) respectively $P_{r} (i), Q_{r} (i)$ (nondegenerate) of Cyclotomic Hecke intertwiners and local power series.

[F1]

The explicit five-branch scalar series and their pair-product/interchange identities are those proved in the explicit normalization and pair-product calculations of Cyclotomic Hecke intertwiners and local power series. Only these scalar identities, not Hecke-side intertwiner relations, are used before establishing the Hecke presentation.

[F2]

The KLR relations (1.6)–(1.15) hold in $R_{α}^{Λ}$ , with (1.6) the cyclotomic relation and (1.12)–(1.13) the dot-sliding relations (The type-A cyclotomic specialization of the QG-KLR algebras); the cyclotomic quotient is finite-dimensional and the $y$ -dots are nilpotent on root strings (Cyclotomic dot annihilators, finite-dimensional blocks, and finite root strings).

[F3]

The type-A Coxeter presentation is the adjacent-transposition presentation of $S_{d}$ (The symmetric group has the Coxeter presentation). Applied to this Coxeter matrix, Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups connects reduced words by braid moves and exposes an adjacent repeat in every nonreduced word using those moves.

Proof

technique · direct

Given: The elements of the Statement and the relations of [F1]–[F2].

1.1F1F2F3algebra

A faithful KLR calculation, before quotient. Take independent dots $u, v, w$ and let $S_{r}$ swap positions, colours and dots. Define the following rational operators locally: equal crossings act by $D = (S - 1) / (u - v)$ and unequal crossings by $S c_{i j}$ , where $c_{i j} = (1 - p) / N_{i j}$ in DG or $(1 - P) / N_{i j}$ in NG. The scalar pair-product identities [F1] give $c_{i j} (u, v) c_{j i} (v, u) = K_{i j} (u, v)$ , with $K = 1, v - u, u - v, - (u - v)^{2}$ . On equal colours direct multiplication gives $D f = (S f) D + (S f - f) / (u - v)$ and $D^{2} = 0$ ; on unequal colours $S c$ simply permutes dot coefficients. Here inverse differences are multiplication on the left, and $c$ is on the right of $S$ . For three distinct colours both braid products have right coefficient $c_{i j} (u, v) c_{i k} (u, w) c_{j k} (v, w)$ at $W = S_{1} S_{2} S_{1}$ . For $(i, i, k)$ both are $(W - S_{1} S_{2}) c_{i k} (u, w) c_{i k} (v, w) / (v - w)$ ; for $(i, k, k)$ both are $(W - S_{2} S_{1}) c_{i k} (u, v) c_{i k} (u, w) / (u - v)$ , with denominators on the left. These two products of coefficients are symmetric in the equal-colour variables, so their divided-difference corrections vanish. For $(i, j, i)$ the longest terms cancel; the identity terms are $- K_{i j} (u, v) / (u - w)$ and $- K_{i j} (w, v) / (u - w)$ , giving difference $(K_{i j} (w, v) - K_{i j} (u, v)) / (u - w) = 0, 1, - 1, u + w - 2 v$ in the four unequal cases. When all colours are equal, both words are $- (1 - S_{1} - S_{2} + S_{1} S_{2} + S_{2} S_{1} - W) / ((u - v) (u - w) (v - w))$ . These explicit operator products verify all KLR mixed, quadratic and braid relations of [F2]. This representation is faithful for formal dot coefficients: moving dots right by the proved sliding rule, use quadratic relations to remove repeated simple reflections and braid relations to reduce each crossing word to a chosen reduced word plus shorter words. This terminates by length (and the finite braid moves of [F3] at each fixed length), giving a spanning family $ψ_{w} e (i)$ . Each braid correction has fewer crossings and each quadratic reduction removes a pair; divided differences of formal series are formal series, so the same reduction works with formal coefficients. Its leading permutation coefficient is a product of nonzero $c_{i j}$ and of $1 / (u - v)$ for equal pairs; proper subwords have smaller permutation length. Maximal-length comparison in the matrix crossed product therefore proves independence of the formal coefficients. Clearing finitely many temporary dot differences is legitimate in this faithful affine calculation. It does not assert faithfulness after taking the cyclotomic quotient.

2.1F1F2step 1.1algebra

The mixed and quadratic computations. In the representation of step 1.1, the displayed inverse Hecke element $ψ N - p$ (DG) or $ψ N - P$ (NG) is literally $S (1 - \hat{p}) - \hat{p}$ or $S (1 - \hat{P}) - \hat{P}$ on every word. On unequal colours this follows by cancellation of the right coefficient $N$ ; on equal colours $ψ = D$ and $D N - 1 = H$ by the equality $H + 1 = (S + 1) (- N / (u - v))$ . For a direct check, write this same inverse operator as $H_{A B} = a_{A B} S + b_{A B}$ with $(h, t) = (1, 0)$ in DG and $(q, q - 1)$ in NG, $b_{A B} = - (A - B)^{- 1}$ in DG and $(1 - q) B / (A - B)$ in NG, and $a_{A B} = h - b_{A B}$ . Substitution gives $b_{A B} + b_{B A} = t$ and $a_{A B} a_{B A} = h + t b_{A B} - b_{A B}^{2}$ ; hence $H^{2} = t H + h$ . Multiplication by $A, B$ gives the mixed relation $H x_{r + 1} - x_{r} H = 1$ in DG, and $H X_{r} H = q X_{r + 1}$ in NG, together with $H^{2} = 1$ or $H^{2} = (q - 1) H + q$ . Alternatively the DG mixed constant on an unequal word is $(i_{r} - i_{r + 1} + u - v) p = 1$ , and on an equal word it is $(u - v) + N = 1$ ; in NG the same spectral substitution gives the stated mixed identity. Commuting dots, distant mixed relations and distant crossings follow because the operators involve disjoint pairs. By step 1.1 these are regular affine-KLR identities and hence evaluate in the nilpotent-dot quotient.

3.1F1step 1.1step 2.1algebra

All five Hecke braid patterns. For a triple with distinct colours, adjacent repeated colours $(i, i, k)$ or $(i, k, k)$ , outer repeated colours $(i, j, i)$ , or all equal colours, the inverse elements of step 2.1 have precisely the same rational Hecke operator $H$ on each successive word. Expansion in the crossed product gives the same six coefficients for both braid words. In the order $1, S_{1}, S_{2}, S_{1} S_{2}, S_{2} S_{1}, S_{1} S_{2} S_{1}$ , they are $b_{A C} (h + b_{A B} b_{B C}), a_{A B} b_{B C} b_{A C}, a_{B C} b_{A B} b_{A C}, a_{A B} a_{A C} b_{B C}, a_{B C} a_{A C} b_{A B}, a_{A B} a_{A C} a_{B C} .$ For example the first word's constant coefficient is $b_{A B}^{2} b_{B C} + a_{A B} a_{B A} b_{A C}$ and its $S_{1}$ coefficient is $a_{A B} (b_{A C} b_{B A} + b_{A B} b_{B C})$ ; the identities below reduce them to the displayed entries. The other entries follow by choosing the permutation terms in two or three factors; swapping $A B$ and $B C$ treats the other word. These formulas apply to all five patterns because $A, B, C$ are independent spectral variables, even when their colour labels coincide. The required scalar reductions are $b_{A B} + b_{B A} = t$ , $a_{A B} a_{B A} = h + t b_{A B} - b_{A B}^{2}$ , and $b_{A B} b_{B C} = b_{A C} (b_{A B} + b_{B C} - t)$ , with $(h, t) = (1, 0)$ or $(q, q - 1)$ . In the outer-repeated pattern the KLR braid correction $u + w - 2 v$ on a double edge was already included in the faithful model of step 1.1, so it is not suppressed by this table. In the equal pattern $D$ accounts for all divided-difference terms. Equality of the six explicit coefficients, followed by faithfulness, proves Hecke braid inside the formal KLR algebra; its regular expressions descend to the cyclotomic quotient. This proves the missing four cases and the all-equal case without assuming any Hecke intertwiner relation in that algebra.

4.1F1F2step 2.1step 3.1construct

The cyclotomic relation and the homomorphism. The cyclotomic relation $\prod_{i} (x_{1} - i)^{λ_{i}} = 0$ (respectively $\prod_{i} (X_{1} - q^{i})^{λ_{i}} = 0$ ) holds because $y_{1}^{(Λ, α_{i_{1}})} e (i) = 0$ is the cyclotomic KLR relation and $x_{1} = \sum_{i} (y_{1} + i_{1}) e (i)$ (respectively $X_{1} = \sum_{i} q^{i_{1}} (1 - y_{1}) e (i)$ ) is a shift of the dot. Hence all defining relations of the affine cyclotomic quotient are satisfied, and the universal property of the presentation gives a unique $F$ -algebra homomorphism $σ ⁣ : H_{d}^{Λ} \to R_{α}^{Λ}$ carrying the generators to the displayed elements.

5.1F2step 4.1

Identification of $σ (e (i))$ . By [F2] the algebra $R_{α}^{Λ}$ is finite-dimensional, so it becomes a finite-dimensional $H_{d}^{Λ}$ -module through $σ$ . The simultaneous generalized weight space for $σ (x_{r}) e (i) = (y_{r} + i_{r}) e (i)$ in DG, or $σ (X_{r}) e (i) = q^{i_{r}} (1 - y_{r}) e (i)$ in NG, on $R_{α}^{Λ}$ is exactly $e (i) R_{α}^{Λ}$ when $i \in I_{α}$ and is $0$ otherwise, by the nilpotence of the dots and the weight relation. Hence $σ (e (i))$ , being the projector onto that generalized weight space for the commuting operators $σ (x_{1}), \dots, σ (x_{d})$ in DG, or $σ (X_{1}), \dots, σ (X_{d})$ in NG, equals $e (i)$ for $i \in I_{α}$ and vanishes otherwise.

6.1step 5.1algebra

Factorization through the sector. Since $e_{α}$ is central and $\sum_{β} e_{β} = 1$ in $H_{d}^{Λ}$ (Central content sectors of the cyclotomic Hecke algebra), and since $σ$ annihilates all idempotents $e (i)$ with $i \notin I_{α}$ by step 5.1, the element $σ (1 - e_{α}) = 0$ . Therefore $σ$ kills $e_{β}$ for $β \neq α$ and every corner $e_{β} H_{d}^{Λ} e_{γ}$ with $(β, γ) \neq (α, α)$ , and it factors through the central summand $H_{α}^{Λ} = e_{α} H_{d}^{Λ} e_{α}$ as a unital homomorphism.

7.1step 4.1step 6.1∎

Conclusion. Steps 1.1, 2.1, 3.1 and 4.1 produce the homomorphism $σ$ from the verified affine and cyclotomic relations; steps 5.1 and 6.1 identify its action on the content idempotents and show that it kills all other sector corners, restricting to a unital homomorphism $H_{α}^{Λ} \to R_{α}^{Λ}$ . No choice principle is used.

Remarks

The scalar normalization is checked before the KLR-to-Hecke calculation. The rational model is faithful only before quotient; the identities evaluated in the cyclotomic algebra contain no inverted equal-colour dot differences.

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Hecke intertwiners produce the local KLR generators

Statement

Let $H_{α}^{Λ}$ be a central content sector, possibly zero, as in Central content sectors of the cyclotomic Hecke algebra, and let $y_{r}$ be the elements of Cyclotomic Hecke intertwiners and local power series. Put $ψ_{r} : = \sum_{i \in I_{α}} φ_{r} q_{r} (i)^{- 1} e (i) (degenerate case), ψ_{r} : = \sum_{i \in I_{α}} Φ_{r} Q_{r} (i)^{- 1} e (i) (nondegenerate case) .$ Then the elements ${e (i) : i \in I_{α}} \cup {y_{1}, \dots, y_{d}} \cup {ψ_{1}, \dots, ψ_{d - 1}}$ of $H_{α}^{Λ}$ satisfy the relations (1.6)–(1.15) of the sign-modified cyclotomic KLR presentation of The type-A cyclotomic specialization of the QG-KLR algebras. Consequently there is a unique $F$ -algebra homomorphism $ρ ⁣ : R_{α}^{Λ} \to H_{α}^{Λ}$ with $e (i) \mapsto e (i)$ , $y_{r} \mapsto y_{r}$ and $ψ_{r} \mapsto ψ_{r}$ .

Facts & Assumptions

Given: The sector $H_{α}^{Λ}$ (Central content sectors of the cyclotomic Hecke algebra), the elements $y_{r}$ , the intertwiners $φ_{r}, Φ_{r}$ and the normalizing series $p_{r} (i), q_{r} (i), P_{r} (i), Q_{r} (i)$ (Cyclotomic Hecke intertwiners and local power series), and the sign-modified KLR presentation (1.6)–(1.15) (The type-A cyclotomic specialization of the QG-KLR algebras).

[F1]

The $y_{r}$ are commuting nilpotents on the sector and the intertwiners satisfy (3.14)–(3.20) in the degenerate case and (4.14)–(4.20) in the nondegenerate case, while the normalizing series satisfy (3.27)–(3.29) and (4.33)–(4.35) with compatibility (3.43), (4.44) (Cyclotomic Hecke intertwiners and local power series).

[F2]

The defining relations of the sector are (3.2)–(3.5) respectively (4.2)–(4.5) together with the cyclotomic relation $\prod_{i} (x_{1} - i)^{λ_{i}} = 0$ respectively $\prod_{i} (X_{1} - q^{i})^{λ_{i}} = 0$ (Affine and cyclotomic type-A Hecke algebras).

Proof

technique · direct

Given: The displayed elements $ψ_{r}$ and the identities of [F1]–[F2].

1.1F1F2algebra

The cyclotomic and dot-sliding relations. Multiply the first-variable cyclotomic relation [F2] by $e (i)$ . For every $j \neq i_{1}$ , the factor $x_{1} - j$ on $e (i)$ in DG has nonzero constant $i_{1} - j$ plus a nilpotent, and the factor $X_{1} - q^{j}$ in NG has nonzero constant $q^{i_{1}} - q^{j}$ plus a nilpotent. Each is therefore invertible in that word corner by the finite geometric series. Cancelling these other-residue factors leaves $(x_{1} - i_{1})^{λ_{i_{1}}} e (i) = 0$ or $(X_{1} - q^{i_{1}})^{λ_{i_{1}}} e (i) = 0$ , respectively. Since these remaining factors are $y_{1}$ and $- q^{i_{1}} y_{1}$ , this is exactly $y_{1}^{λ_{i_{1}}} e (i) = 0$ . If $λ_{i_{1}} = 0$ it says $e (i) = 0$ . No invertibility on the complementary summands is needed. The dot and idempotent relations are immediate from the definitions of the idempotents $e (i)$ and the dots $y_{r}$ , and the dot-sliding relations follow from the dot-sliding intertwiner identities, the definitions of $p_{r} (i), P_{r} (i)$ and the equal-colour case of the normalization.

1.2F1F2algebra

Quadratic and all braid patterns. Work first in the faithful formal affine-Hecke calculation of the normalization supplier, then evaluate its regular identities in the sector. On equal colours the normalized intertwiner is $D = (S - 1) / (u - v)$ , so $D^{2} = 0$ and $D v - u D = 1 = v D - D u$ . On unequal colours it is $S c_{i j} (u, v)$ , with $c = (1 - p) / N$ in DG or $(1 - P) / N$ in NG; its square is $c_{i j} (u, v) c_{j i} (v, u) = K_{i j} (u, v)$ , giving respectively $1, v - u, u - v, (v - u) (u - v)$ . For the braid, put $W = S_{1} S_{2} S_{1}$ and dots $u, v, w$ . Distinct colours give both words $W c_{i j} (u, v) c_{i k} (u, w) c_{j k} (v, w)$ . For $(i, i, k)$ both words are $(W - S_{1} S_{2}) c_{i k} (u, w) c_{i k} (v, w) / (v - w)$ ; for $(i, k, k)$ they are $(W - S_{2} S_{1}) c_{i k} (u, v) c_{i k} (u, w) / (u - v)$ , with denominators on the left and coefficient products on the right. The products are symmetric in the two equal-colour variables, so their divided differences vanish. For $(i, j, i)$ the longest terms cancel and the identity coefficients differ by $(K_{i j} (w, v) - K_{i j} (u, v)) / (u - w)$ ; direct substitution gives $0, 1, - 1, u + w - 2 v$ . On all equal colours the common value is $- (1 - S_{1} - S_{2} + S_{1} S_{2} + S_{2} S_{1} - W) / ((u - v) (u - w) (v - w))$ . These are the supplier's explicit local computations, using only affine-Hecke relations and scalar normalizers, not the opposite relation-map lemma. The double-edge correction and its sign remain valid in characteristic two. Clearing temporary differences in the faithful formal calculation gives regular relations; only those relations, and unit coefficients, are evaluated on nilpotent dots. Hence each of the BK quadratic and braid equations holds in $H_{α}^{Λ}$ .

2.1F1F2step 1.1step 1.2construct

The homomorphism. The verified relations are exactly the presentation of $R_{α}^{Λ}$ recalled in [F2] and in The type-A cyclotomic specialization of the QG-KLR algebras, so the universal property of that presentation gives a unique $F$ -algebra homomorphism $ρ ⁣ : R_{α}^{Λ} \to H_{α}^{Λ}$ sending the generators to the displayed elements. No choice principle is used.

3.1step 1.2step 2.1∎

Conclusion. Steps 1.1 and 1.2 verify all relations of the sign-modified cyclotomic KLR presentation for ${e (i)} \cup {y_{r}} \cup {ψ_{r}}$ , and step 2.1 invokes the universal property to produce $ρ$ . The zero sector is included: all constructions are supported on the projectors $e (i)$ , which may vanish.

Remarks

This is the inverse direction of the Hecke–KLR comparison at the level of a single central content sector; together with The cyclotomic KLR block satisfies the Hecke relations it yields the two mutually inverse maps of the blockwise bridge. The local power-series calculus of Cyclotomic Hecke intertwiners and local power series is the only non-elementary input.

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Cyclotomic Hecke central sectors are isomorphic to type-A cyclotomic KLR algebras

Statement

Over any field $F$ in the charged finite- $e$ conventions, the explicit maps $ρ ⁣ : R_{α}^{Λ} \to H_{α}^{Λ}, σ ⁣ : H_{α}^{Λ} \to R_{α}^{Λ}$ from Hecke intertwiners produce the local KLR generators and The cyclotomic KLR block satisfies the Hecke relations are mutually inverse, for every content $α$ , including zero sectors. Here $H_{α}^{Λ}$ is only the central CRT content sector; no primitivity is assumed. Thus $H_{α}^{Λ} = 0$ if and only if $R_{α}^{Λ} = 0$ , and summing over the finite height- $d$ contents gives $H_{d}^{Λ} ≅ ⨁_{α} R_{α}^{Λ} .$ On idempotents the maps are the literal identity, and the dot image is $x_{r} - i_{r}$ in the degenerate branch and $1 - q_{H}^{- i_{r}} X_{r}$ in the nondegenerate branch. This result precedes the late dimension, basis and primitive-block theorems, and no algebraic-closure hypothesis is required.

Facts & Assumptions

Given: The cyclotomic Hecke algebra $H_{d}^{Λ}$ with its central content sectors ([def-cyclotomic-hecke-central-content-sector-notation]), the specialized cyclotomic KLR algebra $R_{α}^{Λ}$ (The type-A cyclotomic specialization of the QG-KLR algebras), and the two maps constructed by the preceding lemmas (Hecke intertwiners produce the local KLR generators, The cyclotomic KLR block satisfies the Hecke relations).

[F1]

$σ$ carries $e (i)$ to $e (i)$ for $i \in I_{α}$ and kills the idempotents of other contents; it restricts to a unital homomorphism $H_{α}^{Λ} \to R_{α}^{Λ}$ (The cyclotomic KLR block satisfies the Hecke relations).

[F2]

$ρ$ carries $e (i)$ to $e (i)$ , $y_{r}$ to $y_{r}$ and $ψ_{r}$ to the intertwiner quotient $ψ_{r}$ , and is the unique homomorphism with these values (Hecke intertwiners produce the local KLR generators).

[F3]

The content sectors $e_{α}$ are central orthogonal idempotents summing to $1$ (Central content sectors of the cyclotomic Hecke algebra).

Proof

technique · direct

Given: The maps $ρ, σ$ of [F1]–[F2] over the actual field $F$ .

1.1F1F2algebra

The forward composite on dots and crossings. On each KLR word the map $σ$ kills all idempotents outside content $α$ ; in particular, since the inverse polynomial generators of $R$ have exactly the eigenword projectors $e (ν)$ (the nilpotent dots shift distinct scalar eigenvalues), $σ (e_{α}) = 1_{R}$ and $σ$ annihilates every other central sector. The composite $σ ρ$ fixes $y_{r}$ by [F2], and on a crossing $σ (Φ_{r} Q_{r} (i)^{- 1} e (i)) = (ψ_{r} Q_{r} (i) - P_{r} (i) + P_{r} (i)) Q_{r} (i)^{- 1} e (i) = ψ_{r} e (i),$ using the definitions of the local generating elements in Hecke intertwiners produce the local KLR generators; the degenerate case is the same computation with $φ_{r}, p_{r}, q_{r}$ in place of $Φ_{r}, P_{r}, Q_{r}$ . These are literal ordered expressions, not a commutation of $ψ_{r}$ with its coefficient. Hence $σ ρ$ fixes every presentation generator of $R_{α}^{Λ}$ .

1.2F2algebra

The reverse composite on generators. By [F2] the composite $ρ σ$ fixes the idempotents and dots; on a word, $ρ (ψ_{r} Q_{r} (i) - P_{r} (i)) e (i) = Φ_{r} Q_{r} (i)^{- 1} Q_{r} (i) e (i) - P_{r} (i) e (i) = T_{r} e (i)$ in the nondegenerate case, and the same with $s_{r}$ in the degenerate case. Again the evaluation is a finite expression in the nilpotent variables. For the polynomial generators the computations are $ρ σ (x_{r} e_{α}) = \sum_{i} ((x_{r} - i_{r}) + i_{r}) e (i) = x_{r} e_{α}$ in DG and $ρ σ (X_{r} e_{α}) = \sum_{i} q^{i_{r}} (1 - (1 - q^{- i_{r}} X_{r})) e (i) = X_{r} e_{α}$ in NG; their inverses are fixed because homomorphisms preserve inverses. Conversely $σ ρ (y_{r} e (i)) = (σ (x_{r}) - i_{r}) e (i) = y_{r} e (i)$ in DG and $(1 - q^{- i_{r}} σ (X_{r})) e (i) = y_{r} e (i)$ in NG. Along with the projector identities of [F1], these verify every presentation generator of both algebras, including the correct inverse NG substitution. Hence $ρ σ$ fixes every presentation generator of $H_{α}^{Λ}$ .

2.1step 1.1step 1.2algebra

Mutual inverses and the zero-sector statement. Both composites fix all generators of their source algebras, so $ρ σ = id$ and $σ ρ = id$ ; in particular the maps are algebra isomorphisms, and on idempotents they are the literal identity. If $H_{α}^{Λ} = 0$ then the already constructed $σ$ would satisfy $σ (e_{α}) = 1_{R}$ with $e_{α} = 0$ , forcing $1_{R} = 0$ ; conversely the forward identities force $H_{α}^{Λ} = 0$ if $R_{α}^{Λ} = 0$ . Zero rings and rank zero are therefore included without any nonzero-block premise.

3.1F3step 1.1step 2.1∎

Conclusion. Summing the isomorphisms over the finite set of height- $d$ contents and using that the $e_{α}$ are central orthogonal idempotents summing to $1$ by [F3] gives $H_{d}^{Λ} ≅ ⨁_{α} R_{α}^{Λ}$ . Neither exact dimension nor primitive terminology has entered the proof; the full colour-case verifications, affine six-coefficient calculation and equal-colour divided-difference computation reside in the normalization supplier and the two preceding local relation lemmas. No choice principle and no algebraic closure are used.

Remarks

This is the sector-level isomorphism $H_{α}^{Λ} ≅ R_{α}^{Λ}$ for every content, proved before any dimension or primitivity input; it is the algebraic engine of the blockwise bridge. The zero-sector case is handled by the identity $σ (e_{α}) = 1_{R}$ rather than by a nonzero-block hypothesis.

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Basis and dimension of the cyclotomic Hecke algebra

Statement

With the conventions of Affine and cyclotomic type-A Hecke algebras, the images of the ordered monomials $x_{1}^{m_{1}} \dots x_{d}^{m_{d}} w (0 \leq m_{r} < ℓ, w \in S_{d})$ in the degenerate case, respectively $X_{1}^{m_{1}} \dots X_{d}^{m_{d}} T_{w} (0 \leq m_{r} < ℓ, w \in S_{d})$ in the nondegenerate case, form an $F$ -basis of $H_{d}^{Λ}$ . In particular $\dim_{F} H_{d}^{Λ} = ℓ^{d} d!$ and $H_{d}^{Λ}$ is finite dimensional. For $d = 0$ the singleton empty word gives $H_{0}^{Λ} = F$ and dimension one independently of $ℓ$ . If $ℓ = 0$ and $d > 0$ , the relation $1 = 0$ gives the zero algebra and the empty basis; no ambiguous $0^{0}$ convention is used.

Facts & Assumptions

Given: The cyclotomic Hecke algebra $H_{d}^{Λ}$ and its bounded ordered monomials (Bounded ordered monomials span the cyclotomic Hecke quotients), the sector isomorphism (Cyclotomic Hecke central sectors are isomorphic to type-A cyclotomic KLR algebras), and the total dimension of the specialized KLR algebra (Cyclic type-A KLR regular Hom norm gives the total dimension).

[F1]

The bounded ordered monomials span $H_{d}^{Λ}$ : every element is a combination of $X^{a} T_{w}$ (respectively $x^{a} w$ ) with $0 \leq a_{r} < ℓ$ and $w \in S_{d}$ , and there are exactly $ℓ^{d} d!$ such words for $d > 0$ , $ℓ > 0$ (Bounded ordered monomials span the cyclotomic Hecke quotients).

[F2]

For every content $α$ , the sector isomorphism $H_{α}^{Λ} ≅ R_{α}^{Λ}$ holds, so $H_{d}^{Λ} ≅ ⨁_{α} R_{α}^{Λ}$ over the finitely many height- $d$ contents (Cyclotomic Hecke central sectors are isomorphic to type-A cyclotomic KLR algebras).

[F3]

The specialized KLR algebra satisfies $B_{d} = ⨁_{ht β = d} R^{Λ} (β)$ has $\dim_{F} B_{d} = ℓ^{d} d!$ for $d > 0$ , and $B_{0} = F$ (Cyclic type-A KLR regular Hom norm gives the total dimension).

Proof

technique · direct

Given: The data and suppliers of [F1]–[F3]; first assume $d > 0$ and

$ℓ > 0$ .

1.1F1algebra

Exactly $ℓ^{d} d!$ spanning words. By [F1] the bounded ordered monomials span $H_{d}^{Λ}$ , and the number of tuples $(m_{1}, \dots, m_{d}, w)$ with $0 \leq m_{r} < ℓ$ and $w \in S_{d}$ is $ℓ^{d} d!$ . The spanning lemma is stated before, and independent of, the spectral/projector construction, and never imposes the first polynomial independently on every dot.

1.2F2F3algebra

The dimension. By the sector isomorphism [F2] the algebra $H_{d}^{Λ}$ is the direct sum of the finite family of sector algebras $H_{α}^{Λ}$ (one for each content $α$ of height $d$ ), and each $H_{α}^{Λ}$ is isomorphic to the corresponding summand $R_{α}^{Λ}$ of the specialized KLR algebra. Summing dimensions, $\dim_{F} H_{d}^{Λ} = \sum_{ht α = d} \dim_{F} H_{α}^{Λ} = \sum_{ht α = d} \dim_{F} R_{α}^{Λ} = \dim_{F} B_{d} = ℓ^{d} d!,$ the last equality being the total dimension statement of Cyclic type-A KLR regular Hom norm gives the total dimension; grouping the root components $R^{Λ} (β)$ by their content $α$ is legitimate because the idempotents of distinct contents are orthogonal. This is an integer identity, not an equality reduced modulo the characteristic.

2.1step 1.1step 1.2∎

Basis and boundary cases. A spanning family of $ℓ^{d} d!$ vectors in the $ℓ^{d} d!$ -dimensional space $H_{d}^{Λ}$ is a basis, so the bounded ordered monomials form an $F$ -basis and $\dim_{F} H_{d}^{Λ} = ℓ^{d} d!$ . For $d = 0$ the singleton empty word gives $H_{0}^{Λ} = F$ independently of $ℓ$ ; for $ℓ = 0$ and $d > 0$ the imposed relation is $1 = 0$ , the algebra is zero and the empty family is its basis. No ambiguous $0^{0}$ convention is used. The late theorem does not feed the CRT or the early BK construction, and no Frobenius theorem or global highest-weight isometry is used.

Remarks

The proof transfers the independent KLR dimension across the primitive-free sector isomorphism and then uses the bounded spanning family: a spanning family of exactly the dimension is a basis. The level-one case was already proved independently by the explicit JM inverse, and this lemma does not re-derive it.

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Universal cyclotomic Hecke ordered monomials form a basis

Statement

Let $R = Z [z^{\pm 1}, U_{1}^{\pm 1}, \dots, U_{ℓ}^{\pm 1}]$ with $ℓ \geq 1$ , and let $H_{n}$ be the actual nondegenerate cyclotomic presentation of Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules at the first polynomial $\prod_{a} (X_{1} - U_{a})$ . Then $H_{n}$ is a free $R$ -module with indexed $R$ -basis $X^{a} T_{w}, 0 \leq a_{r} < ℓ, w \in S_{n} .$ This basis specializes to every commutative $R$ -algebra. Rank zero is $R$ with the singleton empty word. The separate level-zero positive-rank quotient is zero with the empty family.

Facts & Assumptions

Given: The universal Laurent parameter ring $R$ , the nondegenerate cyclotomic presentation $H_{n}$ with the first-variable polynomial $\prod_{a} (X_{1} - U_{a})$ , and the bounded ordered family $B = {X^{a} T_{w} : 0 \leq a_{r} < ℓ, w \in S_{n}}$ .

[F1]

The images of the bounded ordered monomials span $H_{n}$ over any base ring with invertible parameter: the Bernstein product identities transport the first polynomial to a zero relation with leading coefficient $z^{r - 1} X_{r}^{ℓ}$ and remainders of $r$ -exponent below $ℓ$ involving no variable above $r$ , and reverse-lexicographic induction terminates because the leading coefficients are units of $R$ ; inverse variables are polynomials by the unit constant term of the first polynomial and the invertible finite Hecke generators (Bounded ordered monomials span the cyclotomic Hecke quotients).

[F2]

For a finite cyclic type-A field parameter, the sector isomorphism $H_{α}^{Λ} ≅ R_{α}^{Λ}$ (Cyclotomic Hecke central sectors are isomorphic to type-A cyclotomic KLR algebras) combined with the KLR total dimension gives $\dim_{F} H_{n}^{Λ} = ℓ^{n} n!$ (Cyclic type-A KLR regular Hom norm gives the total dimension).

[F3]

For every prime power $q$ there is a finite field $F_{q}$ , whose multiplicative group is cyclic of order $q - 1$ (For every prime $p$ and $n \geq 1$ , a field with $p^{n}$ elements exists, The multiplicative group $F_{q}^{\times}$ of a finite field is cyclic); the Euler function satisfies $φ (N) \geq \sqrt{N / 2}$ , and a nonzero polynomial over a field has at most its degree roots (Division algorithm for polynomials over a field).

Proof

technique · direct

Given: The universal presentation, the bounded family $B$ , and

the suppliers of [F1]–[F3].

1.1F1algebra

Spanning over $R$ . By [F1] the family $B$ spans $H_{n}$ over the universal ring $R$ : every element is an $R$ -combination of the bounded ordered monomials with $0 \leq a_{r} < ℓ$ and $w \in S_{n}$ . The calculation uses no field division and therefore holds over $R$ ; for $n = 0$ the singleton empty word gives $H_{0} = R$ , and for $ℓ = 0$ , $n > 0$ the quotient is zero with the empty family.

1.2F1F3algebra

Independence by finite-field specialization. Suppose $\sum_{b} f_{b} b = 0$ is a nontrivial relation in $H_{n}$ with $f_{b} \in R$ , not all zero. Fix $b_{0}$ with $f : = f_{b_{0}} \neq 0$ and multiply the relation by a Laurent monomial so that $f \in Z [z, U_{1}, \dots, U_{ℓ}]$ ; choose a prime $p$ for which some nonzero integer coefficient of $f$ survives modulo $p$ , and view $f$ as a polynomial in $U_{1}, \dots, U_{ℓ}$ with coefficients in $F_{p} [z]$ ; let $g (z) \neq 0$ be one of those coefficients. By [F3] the number of primitive elements of $F_{p^{d}}^{\times}$ is $φ (p^{d} - 1)$ , which is unbounded and satisfies $φ (N) \geq \sqrt{N / 2}$ (factor $N$ over prime powers: each factor $φ (r^{a})^{2} / r^{a} = r^{a - 2} (r - 1)^{2} \geq 1$ except $r = 2$ , $a = 1$ , where it is $1 / 2$ ). Choose $d$ with $φ (p^{d} - 1) > \deg g$ and $p^{d} - 1$ larger than every $U$ -degree of $f$ ; then some primitive $z \in F_{p^{d}}^{\times}$ avoids the roots of $g$ . Repeated one-variable root bounds on the nonzero-coordinate grid (each $U_{a}$ is constrained to be a power $z^{c_{a}} \neq 0$ , and the remaining polynomial in the next variable has degree bounded by the $U$ -degree of $f$ ) supply $U_{a} = z^{c_{a}} \neq 0$ with $f (z, U_{1}, \dots, U_{ℓ}) \neq 0$ in $F_{p^{d}}$ . This specialization is exactly an allowed finite cyclic type-A field parameter: $z$ has order $e = p^{d} - 1 \geq 2$ , and the cyclotomic charges $c_{a}$ lie in $Z / e Z$ .

2.1F2step 1.1step 1.2algebra

The specialized basis forces a contradiction. Over the field $F_{p^{d}}$ with the parameter and charges of step 1.2, the early explicit BK maps identify the direct sum of the Hecke central sectors with the standard homogeneous cyclic KLR algebra, whose total dimension is $ℓ^{n} n!$ by [F2]. The bounded spanning of step 1.1 provides exactly $ℓ^{n} n!$ ordered words $X^{a} T_{w}$ , so over the specialized field the family $B$ is a basis; specializing the assumed relation forces $f_{b} (z, U) = 0$ for every $b$ , and in particular $f (z, U) = 0$ , contradicting the choice of $z, U$ in step 1.2. Hence the relation was trivial, the family $B$ is $R$ -linearly independent, and with step 1.1 it is an $R$ -basis of $H_{n}$ .

3.1step 1.1step 2.1∎

Base change and boundary cases. Formation of the presented algebra commutes with arbitrary base change: the presentation is a quotient of a free algebra and its specialized ideal is precisely the specialized relation ideal, so the $R$ -basis $B$ specializes to a basis of $H_{n} \otimes_{R} R^{'}$ for every commutative $R$ -algebra $R^{'}$ (in particular for the LM deformation DVR with $z = q_{H} t$ and $U_{a} = Q_{a} t^{n a}$ ). No integral freeness was inferred from generic rank, no flatness was assumed, and no Schur or primitive-block classification was used.

Remarks

The independence proof is a density argument: algebraic relations over the universal Laurent ring are witnessed in arbitrarily large finite fields with primitive parameter, where the exact dimension $ℓ^{n} n!$ is available from the independent KLR theory. The dependency order is essential: the primitive-free spectral/projector sectors and the explicit field maps precede the field norm, which precedes this universal PBW statement; the universal basis then feeds the local canonical trace and the Murphy/semistandard/Schur constructions. No choice principle is used (the finite fields and roots are chosen by explicit bounded searches).

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The degenerate cyclotomic Hecke center is the symmetric polynomial image

Statement

Let $F$ be any field of characteristic $e > 0$ with $e \geq 2$ , and let $H_{d}^{Λ}$ be the degenerate cyclotomic Hecke algebra with the FIRST relation $f (x_{1}) = \prod_{i \in Z / e Z} (x_{1} - i)^{λ_{i}} = 0$ , $λ_{i} \geq 0$ , $ℓ = \sum_{i} λ_{i} \geq 1$ (Affine and cyclotomic type-A Hecke algebras). Then $Z (H_{d}^{Λ})$ is exactly the image of $F [t_{1}, \dots, t_{d}]^{S_{d}}$ under $t_{r} \mapsto x_{r}$ (Symmetric polynomials as the invariants of variable permutations, The finite symmetric group $S_{n}$ , one-line notation, and cycle notation). The proof is over the original $F$ , for every $d \geq 0$ and level $ℓ$ , and does not divide by $ℓ$ or $d!$ .

Facts & Assumptions

Given: The degenerate affine and cyclotomic presentations, the field $F$ of characteristic $e$ , the level $ℓ$ and the polynomials $f (x_{1}) = \prod_{i} (x_{1} - i)^{λ_{i}}$ .

[F1]

The bounded ordered monomials of the degenerate cyclotomic quotient form a basis (Basis and dimension of the cyclotomic Hecke algebra), and the degree-decreasing transported relation $R_{r} = x_{r}^{ℓ} + \sum c_{a, w} x^{a} w$ with remainders of total degree below $ℓ$ holds (Bounded ordered monomials span the cyclotomic Hecke quotients).

[F2]

The mixed relations are $s_{r} x_{r} = x_{r + 1} s_{r} - 1$ and $s_{r} x_{r + 1} = x_{r} s_{r} + 1$ , with distant mixed commutation; the $s_{r}$ satisfy the Coxeter relations (Affine and cyclotomic type-A Hecke algebras).

Proof

technique · direct

Given: The data of [F1]–[F2]; rank zero is $F$ with symmetric image $F$ ,

so assume $d \geq 1$ .

1.1F1F2algebra

The associated graded. Give $x_{i}$ degree $1$ and the symmetric group degree $0$ . By [F1] the transported relation has leading term $x_{i}^{ℓ}$ with all other dot terms of total degree below $ℓ$ , and the bounded reduction sends monomials of dot degree $\leq r$ to bounded monomials of dot degree $\leq r$ ; independence of the bounded basis makes these the filtration steps $F_{r}$ . The leading mixed relations give the variable permutation action of $S_{d}$ , and the transported relations give $x_{i}^{ℓ} = 0$ in the associated graded. Hence the natural surjection $F [X_{1}, \dots, X_{d}] / (X_{1}^{ℓ}, \dots, X_{d}^{ℓ}) ⋊ F S_{d} ⟶ gr H_{d}^{Λ}$ is an isomorphism: the bounded monomials of the truncated algebra map degree-by-degree onto the basis of [F1].

1.2F2algebra

Symmetric polynomials are central. For $x = x_{i}$ , $y = x_{i + 1}$ , $s = s_{i}$ the relations of [F2] give, by induction on products, $s g (x, y) - g (y, x) s = \frac{g (y, x) - g (x, y)}{x - y}$ for every polynomial $g$ ; the quotient is a polynomial because the numerator vanishes at $x = y$ . Applying this to a polynomial symmetric in $x, y$ shows that it commutes with $s$ , and a globally symmetric polynomial commutes with every $s_{i}$ and every $x_{j}$ , hence is central in $H_{d}^{Λ}$ . No division by an integer or by $x - y$ inside the algebra is used.

2.1step 1.1algebra

Cycle coefficients in the graded centralizer. In $B = F [X_{1}, \dots, X_{d}] / (X_{1}^{ℓ}, \dots, X_{d}^{ℓ}) ⋊ F S_{d}$ , put $h_{r} (I) = \sum X^{b}$ over $0 \leq b_{i} < ℓ$ and $\sum_{i \in I} b_{i} = (∣ I ∣ - 1) (ℓ - 1) + r$ , and $A^{(r)} = h_{r} (I) A$ for a cycle $A$ on $I$ , $0 \leq r < ℓ$ . For a coefficient $g A$ commuting with the $X_{i}$ , the equations are $(X_{i} - X_{A (i)}) g = 0$ . On one cycle they equate coefficients obtained by transferring one unit between consecutive coordinates, with out-of-range coefficients interpreted as zero. At fixed total degree, transfer units until the first $∣ I ∣ - 1$ coordinates are $ℓ - 1$ . If the total is $(∣ I ∣ - 1) (ℓ - 1) + r$ , every coefficient equals the coefficient at $(ℓ - 1, \dots, ℓ - 1, r)$ ; if the total is smaller, the same transfers reach a zero coordinate followed by a transfer to exponent $- 1$ , and all coefficients vanish. Thus the solutions on that cycle have basis $h_{0} (I), \dots, h_{ℓ - 1} (I)$ . For disjoint cycles apply these equations separately to each coefficient in the monomial basis of the complementary variables; this gives exactly the products of disjoint colored cycles as the polynomial centralizer basis. This argument also covers $ℓ = 1$ : the truncated polynomial ring is $F$ , and its centralizer is $F S_{d}$ .

2.2F1F2step 1.1algebra

Reduced-degree powers. Write $f (t) = t^{ℓ} + c_{1} t^{ℓ - 1} + \dots + c_{ℓ}$ . The mixed relations give $x_{i + 1}^{ℓ} = s_{i} x_{i}^{ℓ} s_{i} + \sum_{t = 0}^{ℓ - 1} x_{i}^{t} x_{i + 1}^{ℓ - 1 - t} s_{i}$ . Starting with $x_{1}^{ℓ} = - \sum_{a = 1}^{ℓ} c_{a} x_{1}^{ℓ - a}$ , induction on $i$ shows that $x_{i}^{ℓ}$ has filtration degree at most $ℓ - 1$ and symbol $Y_{i} - c_{1} X_{i}^{ℓ - 1}$ there, where $Y_{i} = \sum_{j < i} (j i)^{(0)}$ . For $k = p ℓ + r$ , $0 \leq r < ℓ$ , it follows that $x_{i}^{k}$ has filtration degree at most $p (ℓ - 1) + r$ and symbol $(Y_{i} - c_{1} X_{i}^{ℓ - 1})^{p} X_{i}^{r}$ . This is a reduced degree: the ordinary degree- $k$ symbol may be zero.

3.1step 2.1algebra

Orbit sums and colored-cycle multiplication. Conjugation permutes the basis of step 2.1, so its fixed vectors have basis the orbit sums $z_{λ}$ , indexed by colored cycle types, including trivial color-zero fixed points to fill rank $d$ . No averaging is used. If $∣ I \cap J ∣ = c > 0$ , direct multiplication gives $h_{r} (I) h_{s} (J) = ℓ^{c - 1} h_{r + s + (c - 1) (ℓ - 1)} (I \cup J)$ . For $c = 1$ , fixing the exponents away from the common variable fixes its exponent uniquely and gives the asserted coefficient one. For $c = 2$ , the basic repeated-edge computation is $h_{0} (i, j)^{2} = ℓ X_{i}^{ℓ - 1} X_{j}^{ℓ - 1}$ : only equal complementary exponents survive, with exactly $ℓ$ choices. Joining the remaining common variables successively reduces the general case to these two computations. Consequently disjoint colored cycles commute; two cycles meeting at one point multiply to their merged cycle of color $r + s$ ; when they meet in at least two points the product vanishes unless $r = s = 0$ , and otherwise is a scalar times disjoint cycles all of color $ℓ - 1$ . Indeed the displayed exponent equals the maximal degree on the union when $c = 2$ , and exceeds it when $c > 2$ and $ℓ > 1$ ; when $ℓ = 1$ every polynomial coefficient is one and the assertion reduces to ordinary cycle multiplication. Iterating gives disjoint-cycle reductions, never increasing the union of supports.

4.1step 3.1step 2.2algebra

The star-cycle expansion. Let $y_{i} (p ℓ + r)$ be the sum of $(j_{1} \dots j_{p} i)^{(r)}$ over ordered, distinct $j_{1}, \dots, j_{p} < i$ , and let $y_{i} (0) = 1$ . Expanding $Y_{i}^{p}$ , sequences with distinct leaves give precisely $y_{i} (p ℓ)$ : the product of the star transpositions is one cycle, with a unique ordered leaf sequence for each such cycle. Any repeated leaf uses at most $p$ vertices including $i$ ; at its first repetition the intersecting cycles have at least two common vertices, so step 3.1 turns the result into saturated (color $ℓ - 1$ ) disjoint cycles or zero. Further star factors preserve saturation by the one-point rule, or vanish by the multiple-point rule. The term $X_{i}^{ℓ - 1}$ is itself a saturated singleton containing $i$ ; choosing it anywhere in $(Y_{i} - c_{1} X_{i}^{ℓ - 1})^{p}$ has the same consequence and uses at most $p$ vertices. Thus this power equals $y_{i} (p ℓ)$ plus saturated disjoint-cycle products on at most $p$ vertices, always with $i$ in their support. Multiplication by $X_{i}^{r}$ gives exactly $y_{i} (p ℓ + r)$ when $r > 0$ , because it raises the color of the distinct-leaf cycle to $r$ , while every saturated error containing $i$ vanishes. (For $ℓ = 1$ there is no $r > 0$ case.) This proves the complete power-symbol formula, including $p = 0$ .

5.1step 1.2step 3.1step 4.1algebra

Triangular symmetric lifts. Index colored cycle types by partitions $μ$ obtained by replacing each cycle of length $a$ and color $r$ by $k = (a - 1) ℓ + r$ and discarding zero parts. These are exactly the partitions satisfying $N (μ) = length (μ) + \sum_{j} ⌊ μ_{j} / ℓ ⌋ \leq d$ ; the inverse adds $d - N (μ)$ trivial fixed points. Put $D (μ) = \sum_{j} (μ_{j} - ⌊ μ_{j} / ℓ ⌋)$ and $p_{μ} = \sum_{ν \sim (μ, 0, \dots, 0)} x_{1}^{ν_{1}} \dots x_{d}^{ν_{d}}$ , summing each distinct exponent tuple once. It is central by step 1.2 and lies in filtration degree $D (μ)$ by step 4.1. Its symbol at that degree is $z_{μ}$ plus orbit sums with strictly smaller $N$ : each product of the leading star cycles on disjoint supports has the prescribed type, and its coefficient is one, since each cycle's largest vertex uniquely chooses its star hub, its ordered other vertices uniquely choose its star expansion, and the unordered exponent tuple chooses each assignment only once, even for repeated parts. Overlapping supports strictly decrease the number of used vertices, as does every error from step 4.1; disjoint-cycle reduction preserves this bound. The entire symbol is central, so the remainder is a combination of orbit sums by step 3.1, with smaller $N$ and the same homogeneous degree. Hence, ordering the finite colored types by $N$ , the symbols of the $p_{μ}$ form a unit triangular change of the orbit-sum basis, degree by degree.

6.1step 1.1step 1.2step 5.1∎

Filtration descent. If $z \in Z (H_{d}^{Λ})$ has filtration degree $D$ , its degree- $D$ symbol lies in $Z (B)$ . By step 5.1 it is a linear combination of degree- $D$ symbols of central symmetric evaluations $p_{μ}$ ; subtract that same combination in $H_{d}^{Λ}$ to lower the filtration degree. Repeating terminates because degrees are nonnegative; in degree zero the same triangular basis applies, and the next filtration step is zero. Thus every central element is a symmetric-polynomial evaluation, and step 1.2 proves the reverse inclusion. Rank zero was handled separately. No division by $ℓ$ , $d!$ or the field characteristic occurs.

Remarks

The symbols used here are taken at the reduced filtration degree of a symmetric evaluation. They need not be polynomials in the degree-one symbols of the $x_{i}$ . For example, at level one with $x_{1} = 0$ , the element $x_{1} + x_{2}$ can have the degree-zero transposition symbol. The colored-cycle construction supplies these symbols explicitly. Exact cyclotomic PBW is a required supplier; while its projectivity and isomorphism prerequisites remain unresolved, this center carrier remains conditional on that input.

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The universal cyclotomic Hecke canonical trace is perfect and symmetric

Statement

For the universal cyclotomic presentation of Universal cyclotomic Hecke ordered monomials form a basis, extract the coefficient of the zero-dot identity-permutation PBW word to define the $R$ -linear map $τ$ . Then $τ (a b) = τ (b a)$ and the pairing $(a, b) \mapsto τ (a b)$ is a perfect $R$ -bilinear associative pairing. Both properties hold after every commutative unit-parameter base change, including the LM deformation DVR. Rank zero is $R$ with $τ (1) = 1_{R}$ . For the separate level-zero positive-rank zero algebra the zero pairing is perfect; no nonzero $τ (1) = 1$ assertion is made.

Facts & Assumptions

Given: The universal parameter ring $R = Z [q_{H}^{\pm 1}, Q_{1}^{\pm 1}, \dots, Q_{ℓ}^{\pm 1}]$ with $ℓ \geq 1$ , the nondegenerate cyclotomic quotient $H_{n}$ with first polynomial $f (X_{1}) = \prod_{a = 1}^{ℓ} (X_{1} - Q_{a})$ , and the PBW basis $B = {X_{1}^{a_{1}} \dots X_{n}^{a_{n}} T_{w} : 0 \leq a_{i} < ℓ, w \in S_{n}}$ (Universal cyclotomic Hecke ordered monomials form a basis).

[F1]

$B$ is an $R$ -basis of $H_{n}$ , valid over every commutative $R$ -algebra (Universal cyclotomic Hecke ordered monomials form a basis); the finite Hecke algebra has standard basis $T_{w}$ with $T_{w} T_{s_{i}} = T_{w s_{i}}$ for increasing length and $q_{H} T_{w s_{i}} + (q_{H} - 1) T_{w}$ for decreasing length (The standard basis of the generic type-A Hecke algebra).

[F2]

The mixed relation can be written $T_{j} X_{j} = X_{j + 1} (T_{j} - q_{H} + 1)$ and gives the Bernstein straightening identity $T_{j} X_{j}^{m} = X_{j + 1}^{m} (T_{j} - q_{H} + 1) - (q_{H} - 1) \sum_{b = 1}^{m - 1} X_{j}^{b} X_{j + 1}^{m - b},$ with $T_{j}^{- 1} = q_{H}^{- 1} (T_{j} - q_{H} + 1)$ (Universal cyclotomic Hecke ordered monomials form a basis).

[F3]

Determinants are multiplicative, and block-triangular determinants multiply over their diagonal blocks (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix, For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ ). For a square matrix over the commutative ring $R$ , cofactor expansion gives $G adj (G) = adj (G) G = (\det G) I$ , so a unit determinant gives an inverse $(\det G)^{- 1} adj (G)$ over $R$ .

Proof

technique · direct

Given: The universal PBW presentation and the data of [F1]–[F3].

1.1F1construct

The trace and its dual basis. Define $τ ⁣ : H_{n} \to R$ by $τ (B) = 1$ at the zero-dot identity word and $0$ at every other PBW basis element, extended $R$ -linearly; by [F1] this is well defined over $R$ and after base change. Put $J_{1} = 1$ and $J_{i} = q_{H}^{- 1} T_{i - 1} J_{i - 1} T_{i - 1}$ in the finite Hecke subalgebra; the $J_{i}$ are units. Define the second family $C = {T_{v} J_{1}^{δ (c_{1})} \dots J_{n}^{δ (c_{n})} X_{1}^{c_{1}} \dots X_{n}^{c_{n}} : 0 \leq c_{i} < ℓ, v \in S_{n}}, δ (0) = 0, δ (c) = - 1 (c > 0) .$ The standard anti-involution fixing the presentation generators fixes each $X_{i}$ , so $B^{*} = {T_{v} X^{c}}$ is a basis; at each fixed profile $c$ , multiplying the finite Hecke factor by the unit $J_{1}^{δ (c_{1})} \dots J_{n}^{δ (c_{n})}$ is an invertible $R$ -linear change of the finite Hecke basis. Hence $C$ is an $R$ -basis of $H_{n}$ .

1.2F1F2algebra

Cyclicity for $X_{1}$ via the tower. At rank one the algebra is commutative, so cyclicity is immediate; for the induction assume $n \geq 2$ . Identify $H_{n - 1}$ with the subalgebra on $X_{1}, \dots, X_{n - 1}, T_{1}, \dots, T_{n - 2}$ ; its bounded words are a subset of $B$ . Put $v_{k} = s_{n - 1} \dots s_{k}$ for $1 \leq k < n$ and $v_{n} = 1$ ; the decomposition $w = u v_{k}$ with $u \in S_{n - 1}$ is length additive, so ${X_{n}^{a} T_{v_{k}} : 0 \leq a < ℓ, 1 \leq k \leq n}$ is a left $H_{n - 1}$ -basis of $H_{n}$ . Define the left-linear projection $E_{n} ⁣ : H_{n} \to H_{n - 1}$ to be $1$ on $X_{n}^{0} T_{v_{n}}$ and $0$ on the other basis elements, and $τ_{n} = τ_{n - 1} \circ E_{n}$ . With $V = T_{n - 1} \dots T_{1}$ and $U = (T_{n - 1} - q_{H} + 1) \dots (T_{1} - q_{H} + 1)$ , the mixed relation gives $V X_{1} = X_{n} U$ , and [F2] gives $V X_{1}^{m} = X_{n}^{m} U + J_{m}$ , where every term of $J_{m}$ has total dot degree $m$ and $X_{n}$ -degree between $1$ and $m - 1$ (the leading branch transfers the full exponent, each correction transfers a strictly smaller positive exponent, and no variable above $n$ occurs). For $m \leq ℓ$ every lower exponent in $J_{m}$ is below $ℓ$ , so $E_{n} (J_{m}) = 0$ ; likewise $E_{n} (X_{n}^{m} U) = 0$ for $1 \leq m < ℓ$ . Multiplying the first-dot relation $f (X_{1}) = 0$ by $V$ gives $0 = X_{n}^{ℓ} U + J_{ℓ} + \sum_{m = 1}^{ℓ - 1} c_{m} (X_{n}^{m} U + J_{m}) + c_{0} V$ , and applying $E_{n}$ yields $E_{n} (X_{n}^{ℓ} U) = 0$ , which completes the missing case $m = ℓ$ . To verify right $X_{1}$ -linearity, test the displayed left tower basis $X_{n}^{a} T_{v_{k}}$ . For $k \geq 2$ , $T_{v_{k}}$ and $X_{n}$ commute with $X_{1}$ ; its projection is zero unless $a = 0, k = n$ , where the equality is immediate. For $k = 1$ , $V X_{1} = X_{n} U$ gives $E_{n} (X_{n}^{a} V X_{1}) = E_{n} (X_{n}^{a + 1} U) = 0$ for $0 \leq a < ℓ$ : bounded positive exponents project to zero, and the endpoint $a + 1 = ℓ$ was just established. Left $H_{n - 1}$ -linearity then extends the equality to every $h$ . Hence $E_{n} (h X_{1}) = E_{n} (h) X_{1}$ for all $h$ , and by left linearity $E_{n} (X_{1} h) = X_{1} E_{n} (h)$ ; induction on $n$ gives $τ_{n} (X_{1} h) = τ_{n} (h X_{1})$ .

1.3F1F2algebra

Cyclicity for the finite generators. Let $p = X^{a}$ be a bounded dot monomial and $w \in S_{n}$ . By [F2], $T_{j} p = (s_{j} p) T_{j} + (1 - q_{H}) X_{j + 1} \frac{p - s_{j} p}{X_{j} - X_{j + 1}};$ for $p = X_{j}^{a} X_{j + 1}^{b}$ times the other variables, the correction monomials have the same total dot degree and both displayed exponents at most $\max (a, b) < ℓ$ (the exponents run as $(a - 1 - k, b + 1 + k)$ or $(a + k, b - k)$ ), so when the exponent profile is positive every term on either side of $τ (T_{j} p T_{w}) = τ (p T_{w} T_{j})$ has a positive bounded profile and evaluates to $0$ . When $p = 1$ , the finite Hecke multiplication formula [F1] gives coefficient of $1$ in $T_{j} T_{w}$ , respectively $T_{w} T_{j}$ , equal to $q_{H}$ if $w = s_{j}$ and $0$ otherwise. This proves $τ (T_{j} h) = τ (h T_{j})$ without using the trace one is proving.

2.1step 1.2step 1.3algebra

Generation and symmetry. The elements $X_{1}, T_{1}, \dots, T_{n - 1}$ generate $H_{n}$ as an $R$ -algebra: the constant term $c_{0} = (- 1)^{ℓ} \prod_{a} Q_{a}$ is a unit, so $f (X_{1}) = 0$ makes $X_{1}^{- 1}$ a polynomial in $X_{1}$ ; each $T_{j}^{- 1} = q_{H}^{- 1} (T_{j} - q_{H} + 1)$ ; and $X_{r} = q_{H}^{1 - r} T_{r - 1} \dots T_{1} X_{1} T_{1} \dots T_{r - 1}$ , with inverses expressed likewise. Moving the generators of a word one at a time around $τ$ , using steps 1.2 and 1.3, gives $τ (a b) = τ (b a)$ for every pair of words, hence for all $a, b$ by $R$ -bilinearity; the pairing $(a, b) \mapsto τ (a b)$ is associative and symmetric.

2.2F1F2step 1.1algebra

Restricted-profile zero values. For $0 \leq j \leq n$ let $U_{j}$ be the span of the PBW words whose dot entries above $j$ vanish, all finite permutations included. It is a right finite-Hecke module. Multiplying its dot part by any polynomial in $X_{1}, \dots, X_{j}$ stays in $U_{j}$ : transport the first cyclotomic relation successively only through $T_{1}, \dots, T_{j - 1}$ , so each resulting relation reduces a power of $X_{i}$ , $i \leq j$ , using variables of index at most $i$ and finite words of indices below $i$ ; repeated Bernstein straightening and the bounded reduction of [F1] therefore never introduce a variable above $j$ . In particular $U_{j}$ is the dot algebra on its first $j$ variables times the finite Hecke algebra. For $1 \leq k < ℓ$ and $h \in U_{j}$ , $τ (X_{j + 1}^{k} h) = 0$ , directly from the bounded basis, since the new exponent is positive and bounded. This statement concerns $h \in U_{j}$ , rather than arbitrary $h \in H_{n}$ .

3.1F1F2step 2.1step 2.2algebra

Transferred restricted-profile powers. Write $q = q_{H}$ . For $1 \leq i \leq j + 1 \leq n$ , $1 \leq k \leq ℓ$ and $h \in U_{j}$ we claim $τ (X_{j + 1}^{k} h) = q^{i - j - 1} τ (T_{j} \dots T_{i} X_{i}^{k} T_{i} \dots T_{j} h)$ , with empty chains when $i = j + 1$ . The mixed relations give $X_{i}^{k} = q^{- 1} T_{i - 1} X_{i - 1}^{k} T_{i - 1} + q^{- 1} (q - 1) \sum_{c = 1}^{k - 1} X_{i}^{c} X_{i - 1}^{k - c} T_{i - 1}$ , by induction on $k$ using that $T_{i - 1}$ commutes with $X_{i - 1} X_{i}$ . To descend from $i$ to $i - 1$ , it suffices to show every correction has trace zero. Put $A = T_{j} \dots T_{i}$ , $B = T_{i} \dots T_{j}$ . For $0 < c < k$ , the mixed relation and the formula $X_{j + 1} = q^{i - j - 1} A X_{i} B$ give the exact identity $A X_{i}^{c} X_{i - 1}^{k - c} T_{i - 1} B h = q^{j - i + 1} A X_{i}^{c - 1} X_{i - 1}^{k - c - 1} T_{i - 1} A^{- 1} X_{j + 1} X_{i - 1} h$ : substitute $B = q^{j - i + 1} X_{i}^{- 1} A^{- 1} X_{j + 1}$ and use $X_{i} T_{i - 1} X_{i}^{- 1} = X_{i - 1}^{- 1} T_{i - 1} X_{i - 1}$ . Expand $X_{i}^{c - 1} X_{i - 1}^{k - c - 1} T_{i - 1} A^{- 1}$ in the reversed PBW order $\sum a_{w, b} T_{w} X^{b}$ using $T_{a}^{- 1} = q^{- 1} (T_{a} + 1 - q)$ . Bernstein straightening swaps the two exponents or replaces them by intermediate exponents, so every exponent stays nonnegative and at most $\max (c - 1, k - c - 1) \leq k - 2$ ; variables have index at most $j + 1$ . Cyclicity from step 2.1 makes the trace of each correction a sum of traces of $X_{j + 1}^{1 + b_{j + 1}} (X_{1}^{b_{1}} \dots X_{j}^{b_{j}}) X_{i - 1} h A T_{w}$ . The parenthesized remaining factor lies in $U_{j}$ by step 2.2, and $1 \leq 1 + b_{j + 1} \leq k - 1 < ℓ$ , so each trace is zero. Downward induction proves the claim, including $k = 1$ , whose correction sum is empty.

4.1step 2.1step 2.2step 3.1algebra

The cyclotomic endpoint. Apply step 3.1 with $i = 1$ , $k = ℓ$ . In $τ (q^{- j} T_{j} \dots T_{1} X_{1}^{ℓ} T_{1} \dots T_{j} h)$ replace $X_{1}^{ℓ}$ using its first cyclotomic relation. Each positive power $X_{1}^{k}$ , $k < ℓ$ , transfers back to $X_{j + 1}^{k}$ by step 3.1 and contributes zero by step 2.2. The constant term contributes $α τ (J_{j + 1} h)$ , where $α = - f (0) = (- 1)^{ℓ + 1} Q_{1} \dots Q_{ℓ}$ and $J_{j + 1} = q^{- j} T_{j} \dots T_{1} T_{1} \dots T_{j}$ . Thus $τ (X_{j + 1}^{ℓ} h) = α τ (h J_{j + 1})$ for every $h \in U_{j}$ . For $j = 0$ this follows directly from the first polynomial; for $ℓ = 1$ there are no positive-power intermediate terms. The restriction on $h$ is essential.

5.1F1F3step 1.1step 2.1step 2.2step 4.1algebra

The full Gram blocks. Let $B_{j}$ and $C_{j}$ be the parts of the two bases with entries above $j$ zero, and let $G_{j} = (τ (b c))$ be their matrix, of size $N_{j} = ℓ^{j} n!$ . For new profiles $b, c$ at position $j + 1$ , cyclicity and commuting dots rewrite an entry as $τ (X_{j + 1}^{b + c} h J_{j + 1}^{δ (c)})$ , where $h = X_{1}^{b_{1} + c_{1}} \dots X_{j}^{b_{j} + c_{j}} T_{w} T_{v} J_{1}^{δ (c_{1})} \dots J_{j}^{δ (c_{j})} \in U_{j}$ by step 2.2; earlier combined exponents may exceed $ℓ - 1$ , and that same reduction argument is why this remains in $U_{j}$ . If $b = c = 0$ the block is $G_{j}$ . If $0 < b + c < ℓ$ it is zero by step 2.2. If $b + c = ℓ$ , then $c > 0$ and step 4.1 cancels the last $J_{j + 1}^{- 1}$ , giving the block $α G_{j}$ . Order rows by $b = 0, 1, \dots, ℓ - 1$ and columns by $c = 0, ℓ - 1, \dots, 1$ . The row and column at zero have only $G_{j}$ at their intersection. Among positive rows and columns the blocks above the diagonal have $b + c < ℓ$ and vanish, the diagonal has $b + c = ℓ$ , and blocks below may be arbitrary. Hence $G_{j + 1}$ is block lower triangular with diagonal $G_{j}, α G_{j}, \dots, α G_{j}$ , so $\det G_{j + 1} = \pm α^{(ℓ - 1) N_{j}} (\det G_{j})^{ℓ}$ . At $j = 0$ , $G_{0} = (τ (T_{w} T_{v}))$ has entry $q^{length (w)} δ_{v, w^{- 1}}$ by [F1], and its determinant is a unit. Induction gives a unit determinant for $G_{n}$ ; since $B$ and $C$ are bases, the pairing identifies $H_{n}$ with its $R$ -dual.

6.1step 1.1step 5.1∎

Base change and boundary cases. Presentation base change commutes with forming $H_{n}$ , and the universal PBW basis is free, so the trace coefficients and the inverse Gram matrix base-change to any commutative ring in which all parameters remain units; in particular they do so for the LM deformation DVR with $q_{H} t$ and $Q_{a} t^{n a}$ . For $n = 0$ use $H_{0} = R$ with $τ (1) = 1_{R}$ ; for $ℓ = 1$ the family $C$ consists of finite $T_{v}$ and the same proof applies. The separate level-zero positive-rank quotient is $0$ , with the zero perfect pairing, and no claim $τ (1) = 1$ is made for it. No generic faithfulness or imported Bremke–Malle trace statement is used.

Remarks

The trace is the canonical coefficient functional on the universal PBW basis; cyclicity is proved by the tower projection and the explicit Bernstein computation, and perfectness by exhibiting the dual basis $C$ with a unimodular triangular Gram change. The scalar $α = (- 1)^{ℓ + 1} Q_{1} \dots Q_{ℓ}$ is forced by the first cyclotomic polynomial. At rank one and level one it is $Q_{1}$ , which fixes the sign independently of a printed source formula. Exact universal PBW is a required input; unresolved prerequisites of that supplier hold this carrier and its consumers. All arguments are finite and use no choice principle.

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The arbitrary-level cyclotomic Hecke Murphy basis is cellular

Statement

For $H_{n}$ in Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules over the universal Laurent parameter ring, the elements $m_{s t}$ indexed by all $ℓ$ -multipartitions of $n$ and standard tableaux $s, t$ of that shape are a basis and a dominance-cellular datum: $m_{s t}^{*} = m_{t s}, m_{s t} h \equiv \sum_{v} r_{(t, v)} (h) m_{s v} (m o d H^{> λ}),$ with coefficients $r_{(t, v)} (h)$ independent of $s$ . The higher spans are actual two-sided ideals. Every commutative specialization with unit parameters inherits the cellular basis; in particular the arbitrary-field finite- $e$ nondegenerate charged specializations ( $q_{H} \neq 1$ ) and the LM deformation DVR retain it. The specialization $z = 1$ here is the multiplicative presentation with $T_{i}^{2} = 1$ , not the additive degenerate cyclotomic Hecke algebra. The universal rank is supplied independently by Universal cyclotomic Hecke ordered monomials form a basis, not by a primitive-block classification.

Facts & Assumptions

Given: The arbitrary-level cyclotomic presentation and its Murphy elements $m_{s t}$ , the universal ordered PBW basis, and the ordinary finite-type-A Murphy cellular basis (Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules, Universal cyclotomic Hecke ordered monomials form a basis, The Murphy tableau elements form a cellular basis).

[F1]

The ordinary Murphy elements form a cellular basis of the finite Hecke algebra with the cell law $m_{s t} h \equiv \sum_{v} r_{(t, v)} (h) m_{s v}$ , with coefficients independent of $s$ , modulo higher ordinary shapes, the standard basis $T_{w}$ exists, and the symmetric group has the Coxeter presentation (The Murphy tableau elements form a cellular basis, The standard basis of the generic type-A Hecke algebra, The symmetric group has the Coxeter presentation).

[F2]

The semistandard Murphy bases and Specht filtrations of the ordinary permutation modules hold, with $# SStd$ multiplicities (Type-A Hecke permutation modules: semistandard Murphy bases and Specht filtrations), and $\sum_{ρ ⊢ m} f_{ρ}^{2} = m!$ (The sum of squares of the standard tableau numbers).

Proof

technique · direct

Given: The multipartition data and the suppliers [F1]–[F2]; write

$N^{\geq λ}$ for the span of all standard $m_{s t}$ with shape at least

$λ$ in dominance, and $N^{> λ}$ for the span of the strictly higher

shapes.

1.1F1algebra

Finite-generator straightening. Every component-preserving row-standard multitableau factors its distinguished permutation into a permutation inside the contiguous component intervals followed by the unique shuffle of the intervals preserving internal order; the two inversion sets are disjoint, so the factorization is length-additive. Multiplying by $u_{λ}$ (which commutes with the Hecke generators internal to the intervals), ordinary Murphy straightening applies in each component; higher ordinary component shapes preserve component sizes and strictly increase multipartition dominance, and the final component shuffle relabels each component by an increasing list whose selected entries at any cutoff are an initial segment, so tableau-prefix inequalities are exactly the ordinary ones at the corresponding initial-segment length. For a fixed initial left row and right multiplication by $T_{i}$ : if $t s_{i}$ is standard the length formula gives the usual standard term (or $q_{H}$ times it plus $(q_{H} - 1) m_{t^{λ} t}$ ); if $i, i + 1$ share a row the result is $q_{H} m$ ; and if they share a component but the interchange fails standardness, factoring out the increasing component shuffle and applying the ordinary cell law gives same-shape terms with the same initial left row and strictly higher multipartition terms, the required transfer identity $T_{c} T_{i} = T_{j} T_{c}$ following from $c s_{i} = s_{j} c$ and the length/quadratic formula. A cross-component column failure is impossible because column standardness is imposed componentwise. Descending induction over a linear extension of multipartition dominance therefore makes each $N^{\geq λ}$ and $N^{> λ}$ stable under all finite generators; multiplying on the left by $T_{d (s)}^{*}$ and applying $*$ makes the higher spans two-sided ideals. This induction concerns spanning sets only and assumes no independence.

1.2F1algebra

The cyclotomic component shift. Let $a = (a_{k})$ with $a_{k} = \sum_{b < k} ∣ λ^{(b)} ∣$ and $u_{a} = u_{λ}$ , and let $b = a + e_{k}$ . Put $u_{b} = u_{a} (X_{a_{k} + 1} - Q_{k})$ ; the factors commute, and for $a = a_{k}$ the defining recurrence gives the exact identity $u_{a} T_{a} \dots T_{1} X_{1} T_{1} \dots T_{a} = q_{H}^{a} (Q_{k} u_{a} + u_{b}),$ by substituting $T_{a} \dots T_{1} X_{1} T_{1} \dots T_{a} = q_{H}^{a} X_{a + 1}$ and adding/subtracting $Q_{k} u_{a}$ ; the case $a = 0$ is included. For $k > 1$ form the multicomposition $ν$ obtained by appending a singleton row to component $k - 1$ and deleting the first node of the first row of component $k$ . Its cumulative sizes are $b$ , so $u_{ν} = u_{b}$ , and its Young subgroup is contained in $S_{λ}$ ; the left length-additive coset expansion gives $x_{λ} u_{b} = \sum_{v} T_{v}^{*} m_{ν}$ . Sorting the rows of $ν$ in each component conjugates its Young subgroup and gives $m_{ν} = T_{d} m_{sort ν} T_{d}^{- 1}$ with $u_{ν}$ commuting with the component-preserving permutation, so the ordinary row-sorting argument expresses each term in standard pairs of shapes at least $sort ν$ , which strictly dominates $λ$ : the cumulative total through component $k - 1$ strictly increases and no later cumulative sum decreases. Hence $x_{λ} u_{b} \in N^{> λ}$ . For $k = 1$ , nonemptiness of component $1$ forces $a_{s} \geq 1$ for all $s \geq 2$ , so $u_{b}$ contains every factor $(X_{1} - Q_{s})$ and vanishes by the cyclotomic relation; again $x_{λ} u_{b} \in N^{> λ}$ .

2.1F1step 1.1step 1.2algebra

Right multiplication by $X_{1}$ . Let $t$ be standard with entry $1$ in component $k$ , an initial node, and initial label $a + 1$ with $a = a_{k}$ . Then $d (t) = (s_{a} \dots s_{1}) w$ with $w$ fixing $1$ and $ℓ (d (t)) = a + ℓ (w)$ : pulling label $a + 1$ to value $1$ crosses exactly the $a$ smaller initial labels, whose final values exceed $1$ . A permutation fixing $1$ has a reduced word in $s_{2}, \dots, s_{n - 1}$ , so $T_{w}$ commutes with $X_{1}$ . Multiplying the identity of step 1.2 on the right by $(T_{1} \dots T_{a})^{- 1}$ and then by $T_{w}$ gives $m_{λ} T_{d (t)} X_{1} = q_{H}^{a} Q_{k} m_{λ} (T_{1} \dots T_{a})^{- 1} T_{w} + q_{H}^{a} x_{λ} u_{b} (T_{1} \dots T_{a})^{- 1} T_{w},$ whose second term lies in $N^{> λ}$ by step 1.2 and finite-generator stability, and whose first term lies modulo $N^{> λ}$ in the fixed-initial-row span. Hence that span is stable under $X_{1}$ ; multiplying on the left by $T_{d (s)}^{*}$ and using the left stability of the higher shapes gives the coefficient-independence law $m_{s t} h \equiv \sum_{v} r_{(t, v)} (h) m_{s v} (m o d N^{> λ})$ first for each generator and then for every word, with coefficients independent of $s$ , and $*$ makes the higher spans two-sided. Since $X_{r} = q_{H}^{1 - r} T_{r - 1} \dots T_{1} X_{1} T_{1} \dots T_{r - 1}$ and the cyclotomic constant term is a unit when the $Q_{a}$ are invertible, these generators generate the Laurent affine quotient.

3.1F2step 2.1algebra

Spanning and cellularity. The smallest multipartition $ω = (\emptyset, \dots, \emptyset, (1^{n}))$ satisfies $u_{ω} = x_{ω} = m_{ω} = 1$ , so the generator-stable span of the standard $m_{s t}$ contains $1$ and is all of $H_{n}$ : the Murphy family spans. By Universal cyclotomic Hecke ordered monomials form a basis, $H_{n}$ is independently free of rank $ℓ^{n} n!$ . Counting standard pairs with the ordinary identity $\sum_{ρ ⊢ m} f_{ρ}^{2} = m!$ at the component sizes $n_{a}$ contributes $(n!)^{2} / \prod_{a} n_{a}!$ , and summing over the multipartition compositions gives $ℓ^{n} n!$ by the multinomial theorem. The spanning family therefore has exactly the independent rank; its coefficient matrix in the ordered PBW basis is square and surjective between free modules, hence invertible, so the family is an $R$ -basis and the proved involution and coefficient-independence laws are actual cellularity. Tensoring the proved basis and presentation supplies every commutative specialization with unit parameters, including repeated charges in the nondegenerate charged branch and the LM deformation DVR; the additive degenerate branch is a different presentation and is not a Laurent-parameter specialization; no generic-flatness inference is made. Rank zero is the singleton empty basis.

4.1step 2.1step 3.1∎

Conclusion. Steps 1.1 and 1.2 establish the generator-stable higher ideals for the finite-generator subalgebra; step 2.1 extends stability to $X_{1}$ and hence to all polynomial generators and gives the coefficient-independence law with two-sided higher spans; and step 3.1 proves that the standard Murphy family is an $R$ -basis of exactly the independent rank, so that the involution and coefficient laws are cellularity in the sense of the Definition. Every commutative specialization with unit parameters inherits the basis and the cell law. No primitive-block classification is used.

Remarks

The theorem is downstream of the universal PBW basis and supplies the cellular/Murphy input of the later Specht, Schur and Jantzen constructions; it does not feed the earlier BK construction. The dominance bookkeeping is explicit: component straightening increases multipartition dominance, and the cyclotomic shift places the correcting terms strictly above the shape.

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Arbitrary-level Murphy Jucys–Murphy action is content triangular

Statement

On the Specht module $S_{λ}$ of Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules, $X_{r}$ acts dominance-triangularly with diagonal $U_{a} z^{j - i}$ at the node $(i, j, a)$ carrying $r$ . Over a field with $z = q_{H}$ of finite order $e \geq 2$ and $U_{a} = q_{H}^{c_{a}}$ , every simultaneous generalized eigenword of $S_{λ}$ is among the standard-tableau residue words $c_{a} + j - i$ modulo $e$ ; every such word has the fixed node-residue multiplicities of $λ$ . The universal triangularity also specializes to the LM deformation DVR and the generic fibre, with no semisimplicity or affine-center premise.

Facts & Assumptions

Given: The arbitrary-level Murphy basis and higher ideals of The arbitrary-level cyclotomic Hecke Murphy basis is cellular, and the ordinary Jucys–Murphy triangularity (The Hecke Jucys-Murphy action is dominance triangular).

[F1]

The Murphy elements $m_{s t}$ form a dominance-cellular basis with higher two-sided ideals $N^{> λ}$ , and $m_{λ} (X_{a + 1} - Q_{k}) = x_{λ} u_{b} \in N^{> λ}$ for $a = a_{k}$ and $λ^{(k)} \neq \emptyset$ (The arbitrary-level cyclotomic Hecke Murphy basis is cellular); the generators satisfy $T_{i} X_{i} = X_{i + 1} T_{i} - (q_{H} - 1) X_{i + 1}$ and $T_{i} X_{i + 1} = X_{i} T_{i} + (q_{H} - 1) X_{i + 1}$ , with distant commutation (Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules).

[F2]

The ordinary multiplicative Jucys–Murphy elements $M_{i} = q_{H}^{1 - i} T_{i - 1} \dots T_{1}^{2} \dots T_{i - 1} = 1 + (q_{H} - 1) L_{i}$ act on ordinary standard tableaux with diagonal $q_{H}^{c o n t e n t}$ (The Hecke Jucys-Murphy action is dominance triangular).

[F3]

Bruhat order on permutations has both the prefix-rank and reduced-subword characterizations, is invariant under inversion, and strictly lowers length on strict intervals (Basic properties of the Bruhat order on $S_{n}$ ). Shift the zero-based convention by one to use labels $1, \dots, n$ .

Proof

technique · direct

Given: The multipartition $λ$ , the Murphy module $S_{λ}$ with its

basis $m_{t}$ , and the data of [F1]–[F2].

1.1F1F2algebra

Initial-row eigenvalues. Consider only a nonempty component $k$ and put $a = a_{k}$ ; empty components contribute no nodes or JM eigenvalues. Then [F1] gives $m_{λ} (X_{a + 1} - Q_{k}) = x_{λ} u_{b} \in N^{> λ}$ . The internal finite Hecke submodule generated by $m_{λ}$ modulo higher global shapes is spanned by standard internal component permutations; each such permutation fixes the smallest internal label $a + 1$ (that label occupies the first node of its component), so its Hecke element has a reduced word in the internal generators with indices at least $a + 2$ and commutes with $X_{a + 1}$ . Hence $X_{a + 1}$ acts as the scalar $Q_{k}$ on that whole submodule, with errors in the two-sided ideal $N^{> λ}$ . The $X$ -recurrence then identifies $X_{a + i}$ on this submodule with $Q_{k}$ times the ordinary multiplicative Jucys–Murphy element $M_{i}$ of [F2], by induction from the finite quadratic relation and the additive-JM recurrence (a polynomial identity also at $q_{H} = 1$ ). Ordinary JM triangularity gives diagonal $Q_{k} q_{H}^{c - r}$ at the node $(r, c, k)$ occupied by local entry $i$ : the internal label $i$ is not a row index, and no extra subtraction of $i - 1$ occurs. For the initial component tableau, which is maximal in tableau dominance, there are no strictly more dominant same-shape terms. Thus for the node $(r, c, k)$ carrying global entry $t$ , $m_{λ} X_{t} \equiv Q_{k} q_{H}^{c - r} m_{λ} (m o d N^{> λ})$ .

2.1F1F3step 1.1algebra

JM triangularity for every standard multitableau. Inducting through a reduced word of $w$ , the relations [F1] give $T_{w} X_{r} = X_{w (r)} T_{w} + \sum_{v < w} \sum_{b} a_{(v, b)} X_{b} T_{v},$ where the corrections are supported on proper Bruhat subwords (at the first correction a generator is deleted, moving its linear $X$ -coefficient through preceding generators deletes further generators, and finite quadratic multiplication preserves Bruhat support). Applying this with $w = d (t)$ and multiplying by $m_{λ}$ on the left, step 1.1 evaluates every surviving $X$ coefficient modulo higher shapes; a correction $m_{λ} T_{v}$ reduces first to a row-standard representative $d (u)$ of its Young-subgroup coset, the removed row subgroup contributing its row eigenvalue, with $d (u) \leq v < d (t)$ ; the Bruhat/tableau-prefix relation on the ordered rows of all components then makes $u$ strictly dominate $t$ , and componentwise straightening of $u$ keeps the shape or produces same-shape tableaux at least $u$ in prefix dominance, hence strictly dominating $t$ . The leading eigenvalue is $Q_{k} q_{H}^{j - i}$ at the node carrying $r$ , because $w (r)$ is its initial-row label under $t = t^{λ} \cdot w = w^{- 1} \cdot t^{λ}$ . Therefore $m_{(t^{λ}, t)} X_{r} \equiv Q_{k} q_{H}^{j - i} m_{(t^{λ}, t)} + \sum_{v ⊲ t} a_{v} m_{(t^{λ}, v)} (m o d N^{> λ}),$ the exact arbitrary-level JM content statement, independent of an affine center theorem, semisimplicity or block classification.

3.1step 2.1algebra∎

Residue words and boundary cases. On the resulting Specht module a total ordering extending tableau dominance triangularizes all the $X_{r}$ simultaneously; the one-dimensional successive quotients have exactly the content characters of step 2.1, so every simultaneous generalized eigenword is among the standard-tableau residue words $c_{a} + j - i (m o d e)$ (The Hecke Jucys-Murphy action is dominance triangular for the ordinary part); the residue multiplicities of every such word equal the node-residue multiplicities of $λ$ , since permuting the labels of a standard tableau does not change the multiset of node residues. For rank zero, $H = F$ , the sole shape and tableau are empty and all assertions have their one-dimensional interpretation. The triangularity is polynomial in the parameters, so it specializes to the LM deformation DVR and to the generic fibre.

Remarks

The lemma supplies the content-triangularity input of the Jantzen and core-reduction theory over the original parameters, with no semisimplicity or affine-center hypothesis; the diagonal is the node content and the corrections are strictly dominant standard tableaux.

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Cyclotomic semistandard row-type sums give permutation-module bases

Statement

Let $H_{n} = H_{n} (R; z; U_{1}, \dots, U_{ℓ})$ be the arbitrary-level cyclotomic Hecke algebra of Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules over a commutative ring $R$ in which $z, U_{1}, \dots, U_{ℓ}$ are units, and let $μ$ be an $ℓ$ -multipartition of $n$ . Then the submodule $M_{μ} = m_{μ} H_{n} \subseteq H_{n}$ has $R$ -basis $m_{(S, t)} = \sum_{s : μ (s) = S} m_{s t},$ where $S$ ranges over the semistandard multitableaux of multipartition shape $λ$ and type $μ$ and $t$ over the standard tableaux of shape $λ$ , with $μ (s)$ the row-and-component reading. The sums have disjoint coefficient- $1$ Murphy pivots, so they are $R$ -independent and $M_{μ}$ is a direct summand of $H_{n}$ as an $R$ -module. The statement holds first over the universal Laurent parameter ring and then over every commutative specialization with unit parameters: the proof constructs, for each $(S, t)$ , an explicit affine element $h$ with $m_{(S, t)} = m_{μ} h$ and uses no quantum-factorial or parameter-difference division, and the equality descends to $R$ by coefficient identities rather than by generic flatness. The Specht filtration of $M_{μ}$ is ordered by higher shapes and contains one copy of the Specht module $S_{λ}$ for each semistandard $S$ , and no primitive-block classification is used.

Facts & Assumptions

Given: The arbitrary-level cyclotomic Hecke algebra and Murphy data of Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules, the affine PBW and localization theory of Universal nondegenerate affine Hecke ordered monomials form a basis, the cellular Murphy basis of The arbitrary-level cyclotomic Hecke Murphy basis is cellular, the finite type-A suppliers Type-A Hecke permutation modules: semistandard Murphy bases and Specht filtrations, The standard basis of the generic type-A Hecke algebra and The symmetric group has the Coxeter presentation, the Bruhat order properties of Basic properties of the Bruhat order on $S_{n}$ .

[F1]

Presentation, multipartitions, initial tableau $t^{λ}$ , the polynomial $u_{λ} = \prod_{k} \prod_{j \leq a_{k}} (X_{j} - U_{k})$ and row sum $x_{λ} = \sum_{w \in S_{λ}} T_{w}$ for the row subgroup $S_{λ}$ , the element $m_{λ} = x_{λ} u_{λ}$ , the right submodule $M_{λ} = m_{λ} H_{n}$ , the involution $*$ , and the Murphy elements $m_{s t} = T_{d (s)}^{*} m_{λ} T_{d (t)}$ ; semistandard multitableaux $S$ of multipartition shape and type, the reading $μ (s)$ , and the sums $m_{(S, t)} = \sum_{s : μ (s) = S} m_{s t}$ (Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules). The flag condition is that a label of component $a$ occurs only in a physical component $b \leq a$ ; the type of $S$ counts each label with the multiplicity of its row, and $# {SStd}_{μ} (λ) = K_{(λ, μ)}$ is the number of such $S$ .

[F2]

The affine algebra $H_{n}^{a f f}$ of [F1] has the two PBW bases $X^{a} T_{w}$ and $T_{w} X^{a}$ , left multiplication by a nonzero Laurent polynomial is injective, and after localizing at the differences $X_{i} - X_{j}$ the algebra maps into the rational crossed product with $Φ_{i} : = T_{i} - c_{i, i + 1}$ , $c_{i j} = (z - 1) X_{j} / (X_{j} - X_{i})$ , satisfying $Φ_{i} f = s_{i} (f) Φ_{i}$ (Universal nondegenerate affine Hecke ordered monomials form a basis).

[F3]

The elements $m_{s t}$ are an $R$ -basis of $H_{n}$ and a dominance-cellular datum with $m_{s t}^{*} = m_{t s}$ and $m_{s t} h \equiv \sum_{v} r_{(t, v)} (h) m_{s v} (m o d H^{> λ})$ , coefficients independent of $s$ ; the higher spans are two-sided ideals and every commutative specialization with unit parameters inherits the basis (The arbitrary-level cyclotomic Hecke Murphy basis is cellular).

[F4]

For the ordinary finite type-A Hecke algebra over any commutative ring with invertible parameter, the permutation module $x_{ρ} H_{n}$ has basis $m_{(S, t)} = \sum_{s : ρ (s) = S} m_{s t}$ over semistandard $S$ of type $ρ$ and standard $t$ , with Specht filtration multiplicity $# {SStd}_{ρ} (λ)$ ; over the generic field the algebra and each row parabolic are split semisimple with simple ordinary Specht modules and the same trivial-character multiplicities, and the identical left row-double-coset identity holds (Type-A Hecke permutation modules: semistandard Murphy bases and Specht filtrations).

[F5]

The $T_{w}$ form a basis of the generic finite type-A Hecke algebra with the multiplication rule $T_{w} T_{i} = T_{w s_{i}}$ for increasing length and $T_{w} T_{i} = (z - 1) T_{w} + z T_{w s_{i}}$ for decreasing length (The standard basis of the generic type-A Hecke algebra).

[F6]

The finite symmetric group has the adjacent-transposition Coxeter presentation (The symmetric group has the Coxeter presentation). The Bruhat subword and prefix-rank characterizations, inversion invariance and length comparison are supplied by Basic properties of the Bruhat order on $S_{n}$ , after shifting the labels by one.

Proof

technique · direct

Given: The data and suppliers of [F1]–[F6]; work first over the universal

Laurent parameter ring and localize only where [F2] permits.

1.1F1F2F6algebra

The affine expansion with only variable-difference denominators. Writing $T_{w}$ as the product of the $Φ_{i}$ and the scalars $c_{i, i + 1}$ along a reduced word of $w$ and moving a Laurent polynomial $f$ leftward across the $Φ$ 's by [F2] expresses $T_{w} f$ as a sum of terms $c_{v} v (f)$ times a monomial in a subword, where $v$ runs over products of subwords of that reduced word, every denominator of $c_{v}$ is a product of differences $X_{i} - X_{j}$ , and each $v$ is Bruhat-below $w$ by the subword characterization of Bruhat order ([F6]). In particular applying this with $f = u_{λ}$ keeps all denominators among the variable differences and introduces no $U_{a} - U_{b}$ , quantum factorial, or Gram determinant.

1.2F1F6algebra

Flagging is Bruhat-downward closed and gives monic divisibility. Let $S$ be semistandard of shape $λ$ and type $μ$ , and for a permutation $σ$ let $π = σ^{- 1}$ map each label to its physical position; set $A_{k} = \sum_{a < k} ∣ λ^{(a)} ∣$ and $B_{k} = \sum_{a < k} ∣ μ^{(a)} ∣$ . By [F1] the flag condition says that a label of component $a$ occupies only a physical component $b \leq a$ , i.e. $# {r \leq B_{k} : π (r) \leq A_{k}} = B_{k}$ for all $k$ . The count is at most $B_{k}$ ; inversion preserves the Bruhat order and a Bruhat-smaller permutation has at least as many entries in every prefix rectangle, so every Bruhat-smaller permutation of a flagged one is flagged. If $σ$ is flagged then $σ (u_{λ}) = \prod_{k} \prod_{j \leq A_{k}} (X_{σ (j)} - U_{k})$ contains the factor $(X_{r} - U_{k})$ for every $r \leq B_{k}$ , because $π (r) \leq A_{k}$ ; hence $σ (u_{λ}) = u_{μ} \prod_{j} \prod_{a (j) < k \leq b (j)} (X_{j} - U_{k})$ with $a (j)$ the physical component of the position occupied by label $j$ and $b (j)$ its type component, a monic polynomial. In particular $u_{μ}$ divides $σ (u_{λ})$ in the Laurent polynomial ring.

1.3F1F3F4F5algebra

The finite action and its flagged restriction. Work over the fraction field $K$ of the universal ring. Put $J_{λ} = \prod_{b} S_{∣ λ^{(b)} ∣}$ , the subgroup preserving the component intervals, and let $V_{b}$ be the ordinary left Murphy cell of shape $λ^{(b)}$ . The shape- $λ$ left cell, restricted to the finite Hecke algebra, is $H_{K} (S_{n}) \otimes_{H_{K} (J_{λ})} ⨂_{b} V_{b}$ . To verify this, factor each standard multitableau into its increasing component shuffle and its internal standard tableaux; the corresponding coset words are length-additive. The map sending the shuffle tensor these internal basis vectors to the corresponding left Murphy vector is a bijection of bases. It intertwines each finite generator: a generator inside a component gives the ordinary cell rule [F4]; a generator exchanging two components gives the two-term standard-word rule [F5]. An internal column failure uses exactly the ordinary Garnir straightening, with its component shuffle factored out. This is the same finite-generator calculation as the cellular proof, so it establishes the tensor model without generic rank inference. Now restrict to the $μ$ -row subgroup $P = S_{μ}$ . Its double cosets with $J_{λ}$ are indexed by matrices $N_{r, b}$ counting labels from $μ$ -row $r$ allocated to physical component $b$ , with row sums the $μ$ -row lengths and column sums $∣ λ^{(b)} ∣$ . In the standard-word basis, $H_{K} (P) T_{d} H_{K} (J_{λ})$ for a minimal double-coset word $d$ is stable under both parabolics. Length-additive multiplication identifies its induced cell subspace with induction from the intersection parabolic; this intersection splits the $r$ -th $μ$ -row into consecutive pieces of sizes $N_{r, b}$ , after the increasing relabelling. Indeed $T_{i} T_{d} = T_{d} T_{j}$ for each paired internal adjacent generator, and the coset words times internal basis vectors give the bijection. Hence the restricted left cell is the direct sum of these allocation subspaces. The flagged left cell consists of exactly the summands for which $N_{r, b} = 0$ whenever $b$ exceeds the type component of row $r$ : every tableau in that summand has precisely this allocation, and the converse is the flag condition. The full left cell includes the unflagged allocations as well.

2.1F1F4step 1.1step 1.2algebra

The left row-double-coset identity and localized membership. Let $s_{0}$ be the least standard refinement of $S$ and $D = S_{μ} d (s_{0})^{- 1} S_{λ}$ . Every row-standard refinement of the semistandard $S$ is standard, and the ordinary length-additive left row-double-coset expansion gives $\sum_{s : μ (s) = S} T_{d (s)}^{*} x_{λ} = \sum_{w \in D} T_{w}$ , each element of $D$ exactly once, with no row-intersection quantum factorial ([F4] proves the identical finite identity; [F1] fixes the elements $T_{d (s)}^{*}$ and $x_{λ}$ ). Since $S_{μ}$ permutes labels inside $μ$ -components and $S_{λ}$ permutes positions inside $λ$ -rows, every $w \in D$ is flagged; hence every Bruhat-subword permutation $v$ in the expansion of step 1.1 is flagged by step 1.2, so $v (u_{λ}) = u_{μ} g_{v}$ with $g_{v}$ a Laurent polynomial, and $T_{w} u_{λ} \in u_{μ} H_{n}^{a f f} [(X_{i} - X_{j})^{- 1}]$ for every $w \in D$ . Writing $b_{0} : = \sum_{w \in D} T_{w}$ and summing, $b_{0} u_{λ} \in u_{μ} H_{n}^{a f f} [(X_{i} - X_{j})^{- 1}]$ .

2.2F3F4F5F6step 1.2algebra

Finite straightening preserves the left flag. Let $W_{μ}$ be the span of the $m_{s t}$ with $s$ $μ$ -flagged, whether or not $μ (s)$ is semistandard. Use the finite-generator calculation in step 1.1 of the cellular supplier [F3], rather than only its cell congruence. For right multiplication by $T_{i}$ , a standard interchange retains $s$ ; a row case is scalar; and a column failure is straightened inside the physical component containing that column. In the last case every left coset word in the ordinary Garnir identity [F4] permutes positions within that component interval, even when its ordinary partition shape changes. Thus before standardization the set of $μ$ -labels in each physical component is unchanged. The increasing component shuffle carries those same sets into the new component diagrams. Quadratic reductions among internal words still permute only those component intervals; replacing a reduced coset word by a Bruhat-smaller word also preserves the flag by step 1.2. Row normalization and further ordinary straightening have the same property. Induction through the finite straightening algorithm therefore retains a flagged left tableau in every term, including higher-shape terms, and proves $W_{μ} T_{i} \subseteq W_{μ}$ . This argument also proves stability under left $T_{i}$ for a $μ$ -row generator: its leading interchange exchanges labels of the same type component, preserving the flag, and its nonstandard terms use the internal straightening just analyzed. Consequently $W_{μ}$ and its intersections with the higher cellular ideals are left modules for the $μ$ -row parabolic.

3.1F1F3F4F5F6step 1.2step 2.2algebra

The exact polynomial-generator correction and its left support. Write $a = a_{k}$ , where $1$ is in component $k$ of $t$ , and $d (t) = (s_{a} \dots s_{1}) w$ with $w$ fixing $1$ and length additivity. Put $P = T_{1} \dots T_{a}$ , with $P = 1$ if $a = 0$ , and $u_{b} = u_{λ} (X_{a + 1} - U_{k})$ . The defining recurrence and distant commutation give the exact equality $m_{s t} X_{1} = z^{a} U_{k} T_{d (s)}^{*} m_{λ} P^{- 1} T_{w} + z^{a} T_{d (s)}^{*} x_{λ} u_{b} P^{- 1} T_{w}$ , the formula proved in steps 1.2 and 2.1 of [F3]. Its first term is in $W_{μ}$ by step 2.2, since $T_{i}^{- 1} = z^{- 1} (T_{i} - (z - 1))$ . For $k = 1$ , $u_{b}$ contains the full cyclotomic product in $X_{1}$ and is zero. For $k > 1$ , let $η$ be the row multicomposition obtained by moving the first node of component $k$ to a new singleton row at the end of component $k - 1$ . Then $u_{η} = u_{b}$ and $S_{η} \subseteq S_{λ}$ , so the exact length-additive left coset expansion is $x_{λ} u_{b} = \sum_{v} T_{v}^{*} m_{η}$ . Attach the labels of $s$ to their original positions before this expansion: the coset representatives choose which node of the first row of component $k$ occupies the new singleton; all other labels stay in their old physical components. Each selected label had type component at least $k$ , so is admissible in component $k - 1$ as well. Sorting the rows of $η$ and straightening in its physical components preserves the resulting component label sets, as in step 2.2. Since $s$ is row-standard, $d (s)$ is a minimal representative for its left row-subgroup coset, so $T_{d (s)}^{*} T_{v}^{*} = T_{(v d (s))^{- 1}}$ is length-additive for these $v \in S_{λ}$ . Each such reduced coset word has the flagged allocation just described; Bruhat-smaller subwords remain flagged by step 1.2, and internal straightening retains its component label sets. Hence every standard left tableau in $T_{d (s)}^{*} x_{λ} u_{b}$ is flagged. Appending $P^{- 1} T_{w}$ preserves this by step 2.2, giving $W_{μ} X_{1} \subseteq W_{μ}$ . The algebra is generated by $X_{1}$ and the $T_{i}$ , and the cyclotomic relation expresses $X_{1}^{- 1}$ as a polynomial in $X_{1}$ , since its constant term is a unit. Thus $W_{μ}$ is a right ideal and $m_{μ} \in W_{μ}$ gives $M_{μ} \subseteq W_{μ}$ . Also $T_{i} m = z m$ for every $m \in M_{μ}$ and every $μ$ -row generator, because $u_{μ}$ commutes with that generator and $T_{i} x_{μ} = z x_{μ}$ .

3.2F1F2step 2.1algebra

Exact unlocalized membership. Expand $b_{0} u_{λ}$ in the left affine PBW basis $X^{a} T_{w}$ of [F2]; its coefficients lie in $P = R^{u n i v} [X_{1}^{\pm 1}, \dots, X_{n}^{\pm 1}]$ . Membership in $u_{μ} H_{n}^{a f f} [(X_{i} - X_{j})^{- 1}]$ from step 2.1 says each coefficient lies in $u_{μ} P [(X_{i} - X_{j})^{- 1}]$ . Every factor $X_{j} - U_{k}$ of $u_{μ}$ is a prime of the unique factorization domain $P$ , is not associated to any difference $X_{i} - X_{j}$ , and is not inverted by the localization, so its multiplicity cannot drop when passing to the localization; hence each coefficient is already divisible by $u_{μ}$ in $P$ . Therefore the PBW coordinates of $u_{μ}^{- 1} b_{0} u_{λ}$ are actual Laurent polynomials and there is an exact unlocalized equality $b_{0} u_{λ} = u_{μ} b$ with $b \in H_{n}^{a f f}$ .

4.1F1F2step 3.2algebra

Eigenvalue descent and the explicit element $h$ . Let $i$ be a generator of the $μ$ -row subgroup, so that $i, i + 1$ lie in one $μ$ -component interval. Expand the double coset into disjoint left cosets, choosing the minimal representative $r$ of each coset. Length additivity gives $b_{0} = \sum_{r} x_{μ} T_{r}$ , so $T_{i} b_{0} = z b_{0}$ by pairing the row-subgroup words under left multiplication by $s_{i}$ ; right multiplication by $u_{λ}$ retains this left eigenvalue, so $T_{i} (u_{μ} b) = z u_{μ} b$ . The element $u_{μ}$ is symmetric in the variables of each $μ$ -interval, so it commutes with $T_{i}$ , and injectivity of left multiplication by the nonzero Laurent polynomial $u_{μ}$ ([F2]) cancels it, giving $T_{i} b = z b$ . Write $b = \sum_{w} T_{w} f_{w}$ in the right PBW basis ([F2]). For each $w$ with $ℓ (s_{i} w) = ℓ (w) + 1$ , comparing coefficients of $T_{s_{i} w}$ in $T_{i} b = z b$ gives $f_{w} + (z - 1) f_{s_{i} w} = z f_{s_{i} w}$ by the quadratic relation $(T_{i} - z) (T_{i} + 1) = 0$ , hence $f_{w} = f_{s_{i} w}$ ; thus $f$ is constant on each left coset $S_{μ} w$ . Taking minimal-length coset representatives $r$ and using the length-additive expansion $x_{μ} T_{r} = \sum_{σ \in S_{μ}} T_{σ r}$ gives $b = x_{μ} h_{0}$ with $h_{0} = \sum_{r} T_{r} f_{r} \in H_{n}^{a f f}$ . Since $x_{μ}$ commutes with $u_{μ}$ , the explicit sum defining $m_{(S, t)}$ becomes $m_{(S, t)} = b_{0} u_{λ} T_{d (t)} = u_{μ} x_{μ} h_{0} T_{d (t)} = m_{μ} h$ with $h = h_{0} T_{d (t)}$ ; this constructs $h$ by finite coset enumeration and polynomial division with no inverted quantum factorial, and the equality is exact, so the Murphy coefficient at the pivot $m_{s_{0} t}$ is $1$ .

5.1F3F4F5step 1.3step 4.1step 2.2step 3.1algebra

The flagged multiplicity upper bound. All finite row parabolics are split semisimple over $K$ by [F4]. In an allocation summand of step 1.3, the multiplicity of their trivial character equals the multiplicity of the trivial intersection character on $⨂_{b} V_{b}$ . This can be seen directly: the finite parabolic symmetrizer divided by its nonzero generic quantum factorial is the trivial-character projector, and its coset expansion identifies the invariant vectors of the induced module with the invariant vectors for the intersection. For each component $b$ put $ρ_{b} = (N_{r, b})_{r}$ , omitting zero parts. By [F4] this intersection multiplicity is $\prod_{b} K_{(λ^{(b)}, ρ_{b})}$ . Summing only over the flag-compatible allocations therefore gives $# {SStd}_{μ} (λ)$ : choose an ordinary semistandard tableau in each component with the prescribed row-label multiplicities and combine them; weak-row and strict-column conditions are componentwise and the allocation supplies exactly the component flag. The cellular-ideal filtration of $W_{μ}$ from steps 2.2 and 3.1 has in each shape the flagged left cell tensored with the full right cell, so exactness of the trivial-character projector gives $\dim_{K} (W_{μ})^{t r i v} = \sum_{λ} # {SStd}_{μ} (λ) # {Std}_{λ}$ . Step 3.1 puts $M_{μ}^{K}$ inside this space; step 4.1 puts all that many independent fibre sums inside $M_{μ}^{K}$ . The two bounds agree, proving that $M_{μ}^{K}$ is their $K$ -span. The generic projector is used only for this dimension bound; the membership identities and subsequent descent divide by no quantum factorial.

6.1F3step 5.1algebra∎

Descent to $R$ , specialization, direct summand, and Specht filtration. Let $h \in m_{μ} H_{R}$ . Its Murphy coordinates in the basis [F3] lie in $R$ , and step 5.1 says that, after extending scalars to $K$ , they vanish outside the admissible fibres and are constant on each admissible fibre; these are literal coefficient identities in $R$ because the Laurent polynomial ring injects into $K$ , so $h$ lies in the span $V_{μ}$ of the $m_{(S, t)}$ . Hence $M_{μ} \subseteq V_{μ}$ , and step 4.1 gives the reverse inclusion, so $M_{μ} = V_{μ}$ over $R$ ; every element of $V_{μ}$ was constructed as $m_{μ} h$ , so this infers no basis from generic flatness. For any commutative specialization $R^{'}$ , the same two inclusions hold after applying $- \otimes_{R} R^{'}$ : each $m_{(S, t)}$ remains in $m_{μ} H_{R^{'}}$ by the explicit identity of step 4.1, and each element of $m_{μ} H_{R^{'}}$ has Murphy coordinates in $V_{μ}$ by the specialized coefficient identities. Finally, with $s_{0}$ the pivot refinement of $S$ , replace in the Murphy basis the vector $m_{s_{0} t}$ by the fibre sum $m_{(S, t)}$ and retain $m_{s t}$ for all non-pivot $(s, t)$ : this change of basis is unitriangular with diagonal entries $1$ over $R$ , so the $m_{(S, t)}$ have disjoint coefficient- $1$ pivots and span an $R$ -direct summand of $H_{n}$ . Ordering the semistandard $S$ in a shape-compatible order, the Murphy cell law [F3] identifies each successive fixed-fibre quotient with the Specht module $S_{λ}$ , one copy for each semistandard $S$ .

Remarks

The lemma supplies the permutation-module basis on which the cyclotomic Schur algebra and the Jantzen theory of this page are built. The localized affine argument is used only to prove explicit divisibility by the monic polynomial $u_{μ}$ ; the geometric statements over $R$ are then coefficient identities in the already proved universal Murphy basis, and no parameter-difference or quantum-factorial division occurs. No choice principle is used: only finite bases, finite coset decompositions, finite filtrations, and explicit polynomial identities occur.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Cyclotomic semistandard sums give the actual Schur endomorphism basis

Statement

For $H_{n}$ , the summands $M_{λ} = m_{λ} H_{n}$ , the direct sum $M = ⨁_{λ} M_{λ}$ and the multipartition-summand algebra $C_{n} = {End}_{H_{n}} (M)$ of Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules, evaluation $f \mapsto f (m_{ν})$ identifies ${Hom}_{H_{n}} (M_{ν}, M_{μ}) ≅ M_{μ} \cap H_{n} m_{ν},$ with $R$ -basis the rectangular sums $m_{(S, T)}$ over $S$ semistandard of shape $λ$ and type $μ$ and $T$ semistandard of shape $λ$ and type $ν$ . Consequently the maps $φ_{S T}$ are actual well-defined endomorphisms of $M$ and form a free dominance-cellular basis of $C_{n}$ with $φ_{S T}^{*} = φ_{T S}$ and $φ_{S T} φ \equiv \sum_{W} r_{(T, W)} (φ) φ_{S W} (m o d C_{n}^{⊳ λ})$ with coefficients independent of $S$ . The formation of this basis and its constants specializes to every commutative unit-parameter base, including the LM deformation DVR. The proof uses the perfect symmetric canonical trace (The universal cyclotomic Hecke canonical trace is perfect and symmetric) and the codomain direct-summand pivots (Cyclotomic semistandard row-type sums give permutation-module bases); no field-only double-annihilator and no nonsaturated-lattice argument is used.

Facts & Assumptions

Given: The Data of [F1], the cellular Murphy basis [F2], the permutation module basis [F3], the canonical trace [F4], and the cellular-algebra definition [F5].

[F1]

$H_{n}$ , the submodules $M_{λ} = m_{λ} H_{n}$ , the Murphy elements $m_{s t}$ , the elements $m_{(S, t)}$ and $m_{(S, T)}$ , the direct sum $M = ⨁_{λ} M_{λ}$ and $C_{n} = {End}_{H_{n}} (M)$ , the presentation involution $*$ , and the definition of $φ_{S T}$ as the map sending $m_{α} h$ to $δ_{α, ν} m_{(S, T)} h$ (Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules).

[F2]

The $m_{s t}$ are an $R$ -basis of $H_{n}$ with $m_{s t} h \equiv \sum_{v} r_{(t, v)} (h) m_{s v} (m o d H^{> λ})$ , coefficients independent of $s$ ; the higher spans are two-sided ideals, and the basis and law specialize to every commutative unit-parameter base (The arbitrary-level cyclotomic Hecke Murphy basis is cellular).

[F3]

$M_{μ} = m_{μ} H_{n}$ has $R$ -basis the sums $m_{(S, t)} = \sum_{s : μ (s) = S} m_{s t}$ over semistandard $S$ of shape $λ$ and type $μ$ and standard $t$ , with disjoint coefficient- $1$ Murphy pivots, and is a direct summand of the codomain $H_{n}$ (Cyclotomic semistandard row-type sums give permutation-module bases).

[F4]

The canonical coefficient trace $τ$ satisfies $τ (a b) = τ (b a)$ and makes $(a, b) \mapsto τ (a b)$ a perfect associative $R$ -bilinear pairing, over the universal ring and after every commutative unit-parameter base change (The universal cyclotomic Hecke canonical trace is perfect and symmetric).

[F5]

A cellular basis ${c_{s t}^{λ}}$ of an $R$ -algebra $A$ requires an involutory anti-automorphism $i$ with $i (c_{s t}^{λ}) = c_{t s}^{λ}$ and the triangular multiplication $c_{s t}^{λ} a \equiv \sum_{v} r_{v} c_{s v}^{λ} (m o d A^{⊳ λ})$ with $r_{v}$ independent of $s$ (Cellular bases and cell modules).

Proof

technique · direct

Given: The data and suppliers of [F1]–[F5], over the universal Laurent

parameter ring.

1.1F3F4algebra

Annihilator lemma. Let $m \in H_{n}$ be such that $m H_{n}$ is a direct summand of the codomain $H_{n}$ , put $K = rAnn (m) = {a : m a = 0}$ and let $L_{m} (a) = m a$ . The functional $f$ on $H_{n}$ vanishes on $K$ exactly when $f = g \circ L_{m}$ for some functional $g$ on $H_{n}$ : one direction is immediate, and for the other $f$ descends to the split quotient $H_{n} / K ≅ m H_{n}$ and extends to $H_{n}$ by zero on a complementary summand. Hence, for $x$ with $τ (x k) = 0$ for all $k \in K$ , the functional $a \mapsto τ (x a)$ vanishes on $K$ , so $τ (x a) = g (m a) = τ (y m a)$ for the element $y$ representing $g$ under the perfect pairing [F4], and $τ ((x - y m) a) = 0$ for all $a$ , so $x = y m \in H_{n} m$ . Since $K$ is a right ideal, the converse also holds: if $x = y m$ then $τ (x k) = τ (y m k) = 0$ for $k \in K$ , and if $τ (x k) = 0$ for all $k \in K$ then $τ ((x k) a) = 0$ for all $a$ because $k a \in K$ , so perfectness gives $x k = 0$ . Therefore $lAnn (rAnn (m)) = H_{n} m$ .

2.1F1F3step 1.1algebra

Evaluation is a bijection onto the intersection. The cyclic module $M_{ν} = m_{ν} H_{n}$ [F1] makes evaluation $f \mapsto f (m_{ν})$ injective on ${Hom}_{H_{n}} (M_{ν}, M_{μ})$ , since $f (m_{ν} a) = f (m_{ν}) a$ . Its image is exactly ${x \in m_{μ} H_{n} : x K = 0}$ with $K = rAnn (m_{ν})$ : the inverse sends such an $x$ to the well-defined map $m_{ν} a \mapsto x a$ , which is a right $H_{n}$ -map, and well-definedness is precisely $x K = 0$ . By step 1.1 with $m = m_{ν}$ — whose image is a codomain direct summand by [F3] — this image is $m_{μ} H_{n} \cap H_{n} m_{ν}$ , so ${Hom}_{H_{n}} (M_{ν}, M_{μ}) ≅ M_{μ} \cap H_{n} m_{ν}$ .

3.1F2F3step 2.1algebra

The intersection basis. By [F3] the module $M_{μ}$ has basis $m_{(S, t)}$ over semistandard $S$ of type $μ$ and standard $t$ , whose Murphy coordinates are constant on each admissible left fibre and zero off them ([F2]); applying the involution $*$ to the basis of $M_{ν}$ gives a basis of $H_{n} m_{ν}$ of the elements $m_{(s, T)}$ over semistandard $T$ of type $ν$ and standard $s$ , whose Murphy coordinates are constant on each admissible right fibre and zero off them. Hence an element of the intersection has Murphy coordinates that are zero unless the left fibre is admissible and the right fibre is admissible, and constant on each simultaneous pair of fibres. The rectangular sums $m_{(S, T)}$ have exactly these coordinate properties, with disjoint coefficient- $1$ pivots at the pairs $(s_{0}, T_{0})$ of pivot refinements, and conversely every such coordinate family is the corresponding $R$ -linear combination. Therefore the $m_{(S, T)}$ are an $R$ -basis of $M_{μ} \cap H_{n} m_{ν}$ .

4.1F1step 3.1step 2.1algebra

The endomorphism basis. For $S, T$ of common shape $λ$ and types $μ, ν$ , formula $φ_{S T} (m_{α} h) = δ_{α, ν} m_{(S, T)} h$ is well-defined: the right side depends on $m_{α} h \in M_{α}$ only through $h$ modulo $rAnn (m_{α})$ , and for $α \neq ν$ the map is defined to be zero, while for $α = ν$ well-definedness is exactly the coordinate property of step 3.1; multiplicativity in $h$ makes each $φ_{S T}$ an $H_{n}$ -map, and every endomorphism of $M$ is determined by its restrictions to the finitely many cyclic summands. Evaluation on each $M_{ν}$ delivers by step 2.1 the whole intersection $M_{μ} \cap H_{n} m_{ν}$ , so the $φ_{S T}$ form an $R$ -basis of $C_{n} = {End}_{H_{n}} (M)$ , finite and free.

5.1F1F2F5step 4.1algebra

Cellular laws. Define an $R$ -linear involution on the established endomorphism basis by $φ_{S T}^{*} = φ_{T S}$ . We verify anti-multiplicativity by evaluation; no generator-fixing star map on the right-module direct sum is assumed. If $m_{(S, T)} = m_{μ} h_{S T}$ then $m_{(T, S)} = h_{S T}^{*} m_{μ}$ , and for a composable pair with $type (T) = type (U)$ the star of the evaluated product $m_{(S, T)} h_{U V}$ is $h_{U V}^{*} m_{(T, S)} = m_{(V, U)} h_{T S}$ , while for non-composable types both products vanish; evaluation injectivity (step 4.1) gives $(φ_{S T} φ_{U V})^{*} = φ_{V U} φ_{T S}$ , hence $φ_{S T}^{*} = φ_{T S}$ . For the multiplication rule, evaluate $φ_{S T} φ_{U V}$ at the cyclic generator $m_{type (V)}$ in its source summand: it is $m_{(S, T)} h_{U V}$ when $type (U) = type (T)$ and zero otherwise. The Murphy cell law [F2] expands this same-shape part as $\sum_{s \in fibre S} \sum_{v} b_{(T, v)} (h_{U V}) m_{s v}$ with $b_{(T, v)} = \sum_{t \in fibre T} r_{(t, v)} (h_{U V})$ independent of $s$ and $S$ ; because the product is an actual $H_{n}$ -map and step 3.1 groups $v$ into its semistandard fibres, the coefficients are constant on each right fibre, so the coefficient of $φ_{S W}$ depends on $T, W$ and the multiplier only, and all remaining terms have strictly higher shape. Thus the datum ${φ_{S T}}$ with the dominance poset on shapes satisfies [F5]: it is a free dominance-cellular basis of $C_{n}$ .

6.1F2F4step 5.1algebra∎

Specialization. Every step above is an identity of coefficients in the universal Laurent ring with denominators inverted only when they are units: the trace, the annihilator computation, the two one-sided bases, the intersection basis, and the structure constants all live over the universal ring and involve no field, dimension count, rationalization, or generic flatness. Hence applying any commutative base change in which the parameters are units — including the LM deformation DVR and arbitrary fields — preserves the well-definedness, the basis, the transpose rule, and the cellular multiplication constants.

Remarks

The lemma closes the well-definedness obligation of the definition of $C_{n}$ and of the maps $φ_{S T}$ : the double-annihilator identity is derived from the perfect canonical trace together with the codomain direct-summand property of $m_{ν} H_{n}$ , and the intersection basis is derived from the one-sided semistandard bases, so no field-only double-annihilator or nonsaturated-lattice argument is used. The intersection basis groups the two admissible Murphy coordinate fibres independently. No choice principle is used: all bases and coset sums are finite.

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Cyclotomic Schur standards and corner functor control block linkage

Statement

For $C_{n} = {End}_{H_{n}} (⨁_{λ} m_{λ} H_{n})$ of Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules over a field $F$ with unit parameters, its standards $Δ_{λ}$ have nonzero absolutely simple heads $L_{λ}$ indexed by all multipartitions, with $[Δ_{λ} : L_{λ}] = 1$ and $[Δ_{λ} : L_{μ}] \neq 0$ only if $λ ⊵ μ$ ; the algebra $C_{n}$ is split quasihereditary. The standard classes form a $Z$ -basis of $G_{0} (C_{n} -mod)$ , and the weight dimensions $\dim Δ_{λ} e_{ν} = K_{(λ, ν)}$ form a unitriangular multipartition Kostka matrix. Over the universal ring and the LM deformation DVR the standards and their forms are free and the weight idempotents satisfy $e_{ν}^{*} = e_{ν}$ . The exact corner functor $V \mapsto V e_{ω}$ gives $Δ_{λ} e_{ω} ≅ S_{λ}$ and $L_{λ} e_{ω} ≅ D_{λ}$ (possibly $0$ ), so that $[Δ_{λ} : L_{μ}] = [S_{λ} : D_{μ}]$ whenever $D_{μ} \neq 0$ . The centers $Z (C_{n})$ and $Z (H_{n})$ are canonically isomorphic because $M_{ω} = H_{n}$ is a summand, and $Δ_{λ}, Δ_{μ}$ share a $C_{n}$ -block if and only if $S_{λ}, S_{μ}$ share an $H_{n}$ -block; standard and Specht linkage is exactly shared-simple-factor linkage. No equality-of-residue-contents-to-primitive-block assertion is made.

Facts & Assumptions

Given: The Schur datum of [F1], the semistandard endomorphism basis [F2], the finite cellular-head theory [F3], the block and Grothendieck conventions [F4]–[F5], and the Jucys–Murphy content triangularity [F6].

[F1]

The algebra $H_{n}$ , the summands $M_{λ} = m_{λ} H_{n}$ , the direct sum $M = ⨁_{λ} M_{λ}$ with $ω = (\emptyset, \dots, \emptyset, (1^{n}))$ and $m_{ω} = 1$ , the endomorphism algebra $C_{n} = {End}_{H_{n}} (M)$ with ordinary composition, the presentation involution $*$ , and the weight projections $e_{ν}$ onto $M_{ν}$ (Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules).

[F2]

The maps $φ_{S T}$ are a free dominance-cellular basis of $C_{n}$ with $φ_{S T}^{*} = φ_{T S}$ and $φ_{S T} φ \equiv \sum_{W} r_{(T, W)} (φ) φ_{S W} (m o d C_{n}^{⊳ λ})$ , coefficients independent of $S$ (Cyclotomic semistandard sums give the actual Schur endomorphism basis).

[F3]

For a finite-dimensional cellular algebra over a field with cell modules $C_{λ}$ and quotients $D_{λ} = C_{λ} / R_{λ}$ : every nonzero $D_{λ}$ is an absolutely simple head, all simple modules occur exactly once, $[C_{λ} : D_{μ}] \neq 0 \Rightarrow λ \geq μ$ and $[C_{μ} : D_{μ}] = 1$ ; every nonzero cell module has all composition factors in one block, and two nonzero cell modules share a block exactly when linked by a finite chain of nonzero cell modules with neighbouring terms sharing a simple factor (Nonzero-form cellular heads, cell purity and shared-factor linkage).

[F4]

Blocks of a finite-dimensional algebra are the primitive-central-idempotent blocks (Blocks of a finite-dimensional algebra via primitive central idempotents).

[F5]

$G_{0}$ of a finite-dimensional module category with the Jordan–Hölder theorem: composition classes form a $Z$ -basis (Grothendieck group of an essentially small abelian category, Jordan–Hölder theorem for modules).

[F6]

$X_{r}$ acts dominance-triangularly on $S_{λ}$ with diagonal $U_{a} z^{j - i}$ at the node carrying $r$ (Arbitrary-level Murphy Jucys–Murphy action is content triangular).

Proof

technique · direct

Given: The data and suppliers of [F1]–[F6].

1.1F1F4algebra

Centers and blocks of an endomorphism algebra containing the regular module. Let $M$ be a finite right $H$ -module containing $H$ as a direct summand with projection $e$ , and $C = {End}_{H} (M)$ . Then $e C e = {End}_{H} (H) ≅ H$ by left multiplication, and $C e ≅ {Hom}_{H} (H, M) ≅ M$ by evaluation at $1$ . If $c \in Z (C)$ , then $c$ restricts on the $H$ -summand to left multiplication by some $h$ ; for each $m \in M$ the $H$ -map $f_{m} : H \to M$ , $a \mapsto m a$ , is the off-diagonal block $j f_{m} π \in C$ through $H$ , and commuting it with $c$ and evaluating at $1$ gives $c (m) = m h$ . Since $c$ is $H$ -linear, $c (a b) = c (a) b$ for $a, b \in H$ gives $a b h = a h b$ , so with $a = 1$ we get $h b = b h$ and $h \in Z (H)$ . Conversely every $h \in Z (H)$ defines the $H$ -linear endomorphism $m \mapsto m h$ , which commutes with every $H$ -endomorphism of $M$ , hence is central in $C$ . These constructions are inverse unital algebra maps, so $Z (C) ≅ Z (H)$ and primitive central idempotents correspond bijectively ([F4]). For $M = ⨁_{λ} M_{λ}$ we have $M_{ω} = H_{n}$ , so this applies with $e = e_{ω}$ .

1.2F1F2F3algebra

Schur standards and nonzero heads. By [F2] the right cell module $Δ_{λ}$ of $C_{n}$ has basis the $v_{T}$ over semistandard $T$ of shape $λ$ . The unique tableau $T^{λ}$ of shape and type $λ$ gives $φ_{(T^{λ}, T^{λ})} = e_{λ}$ , the identity projection onto $M_{λ}$ , an actual idempotent; evaluating the cell form at $v_{T^{λ}}$ gives $⟨ v_{T^{λ}}, v_{T^{λ}} ⟩ = 1$ , so every standard is nonzero with nonzero form. Applying the finite cellular-head theory [F3] to $C_{n}$ (finite-dimensional and cellular over $F$ by [F2]) shows every head $L_{λ} = Δ_{λ} / r a d Δ_{λ}$ is nonzero, absolutely simple, the $L_{λ}$ are pairwise distinct and exhaust the simple $C_{n}$ -modules, and $[Δ_{λ} : L_{μ}] \neq 0$ implies $λ ⊵ μ$ with $[Δ_{λ} : L_{λ}] = 1$ .

1.3F2F3algebra

Split quasiheredity by an explicit heredity chain. Order the multipartitions in a linear extension of dominance and remove the maximal remaining cell layers successively; in the relevant quotient algebra $B$ the next layer $J$ has basis $c_{s t}$ with a distinguished index $u$ whose Gram value is $G (u, u) = 1$ . Then $e = c_{u u}$ is idempotent, $c_{s u} c_{u t} = c_{s t}$ , and the multiplication map $B e \otimes_{F} e B \to J$ is an isomorphism on these basis elements; also $e B e = F e$ , since the cell law and maximality of $J$ show every $e b e$ is a scalar multiple of $e$ . Hence $B e$ is projective, $J$ is a direct sum of copies of $B e$ , and $e r a d (B) e = 0$ because the corner is a field. Thus $J$ is a split heredity ideal, and iterating gives a split heredity chain whose standards are the cell modules $Δ_{λ}$ ; so $C_{n}$ is split quasihereditary over $F$ .

1.4F1F2F6algebra

Weight dimensions and the unitriangular Kostka matrix. The weight projections $e_{ν} = φ_{(T^{ν}, T^{ν})}$ are self-adjoint because $φ_{S T}^{*} = φ_{T S}$ ([F2]), and the cell action gives $v_{T} e_{ν} = δ_{type (T), ν} v_{T}$ , so $\dim Δ_{λ} e_{ν}$ is the number $K_{(λ, ν)}$ of semistandard tableaux of shape $λ$ and type $ν$ ; this already shows the standards and their forms are defined over the universal ring, free on the $v_{T}$ . A semistandard tableau of shape $λ$ and type $ν$ forces $λ ⊵ ν$ : for a component $k$ and row cutoff $r$ , labels of type components below $k$ cannot occupy physical components at least $k$ , labels of type component $k$ with row at most $r$ cannot occupy physical components greater than $k$ and occupy at most $r$ nodes per column of physical component $k$ by strict column increase, so the count $\sum_{a < k} ∣ ν^{(a)} ∣ + \sum_{i \leq r} ν_{i}^{(k)}$ is at most $\sum_{a < k} ∣ λ^{(a)} ∣ + \sum_{i \leq r} λ_{i}^{(k)}$ , which is exactly the multipartition dominance inequality; and $T^{λ}$ is the unique tableau of shape and type $λ$ , so the Kostka matrix is unitriangular.

2.1F3F5step 1.2algebra

The standard basis of $G_{0}$ and block purity. The matrix of standard multiplicities is unitriangular with diagonal $1$ by step 1.2, so inverting it by backward substitution shows that the classes $[Δ_{λ}]$ form a $Z$ -basis of $G_{0} (C_{n} -mod)$ ([F5]). Projecting to blocks is exact. Every $Δ_{λ}$ is nonzero and block-pure by [F3], so if a class supported in one block equals $\sum_{λ} a_{λ} [Δ_{λ}]$ , then every coefficient of a standard lying in another block vanishes by this basis independence; this is the block-separation input used by the Jantzen equivalence.

2.2F1F2F3step 1.1algebra

The corner functor and the radical comparison. Put $e = e_{ω}$ ; then $e C e ≅ H_{n}$ by step 1.1, and $V \mapsto V e$ is exact. Semistandard tableaux of type $ω$ are canonically standard multitableaux: replacing the label $(r, ℓ)$ by $r$ is a bijection onto the standard tableaux, and the surviving standard vectors $v_{t}$ of $Δ_{λ} e$ carry the same right- $H_{n}$ action as the Murphy vectors $m_{t}$ of $S_{λ}$ by the cell multiplication [F2]; this is an explicit isomorphism $Δ_{λ} e ≅ S_{λ}$ . The forms also agree: in $C_{n}$ , $⟨ v_{s}, v_{t} ⟩ e_{λ} \equiv φ_{(T^{λ}, s)} φ_{(t, T^{λ})}$ modulo higher $C_{n}$ -shapes, and evaluating the two factors at $m_{λ}$ gives the product $m_{(t^{λ}, s)} m_{(t, t^{λ})}$ , whose coefficient of $m_{λ}$ modulo higher $H_{n}$ -shapes is exactly the Specht form entry; a higher Schur map evaluates to a sum of higher Murphy shapes by its explicit definition, so no shape error is lost. Since $e^{*} = e$ , the $e$ and $1 - e$ parts of $Δ_{λ}$ are orthogonal: if $x$ pairs to zero against $Δ_{λ} e$ , then $⟨ x, y (1 - e) ⟩ = ⟨ x (1 - e), y ⟩ = 0$ , so $r a d (Δ_{λ}) e = r a d (Δ_{λ}) \cap Δ_{λ} e = r a d (Δ_{λ} e)$ . Hence $L_{λ} e ≅ D_{λ}$ for every $λ$ , with $D_{λ}$ allowed to be zero. Applying the exact functor to a composition series of $Δ_{λ}$ , each simple factor $L_{μ}$ becomes $D_{μ}$ or $0$ , and the nonzero $D_{μ}$ are pairwise distinct and exhaustive by [F3]; hence $[Δ_{λ} : L_{μ}] = [S_{λ} : D_{μ}]$ for every $μ$ with $D_{μ} \neq 0$ , with no simple-label bijection through $e$ assumed.

3.1F3step 1.1step 2.2algebra∎

Typed Schur/Specht block linkage. The center isomorphism of step 1.1 sends a central block idempotent $b$ of $H_{n}$ to the same right-action projection on $M$ , hence to the corresponding central idempotent $b_{C}$ of $C_{n}$ , and on the corner $V b_{C} e = (V e) b$ . Every $Δ_{λ}$ is nonzero and block-pure by [F3], and $Δ_{λ} e ≅ S_{λ}$ is nonzero because $S_{λ}$ has its Murphy standard-tableau basis; consequently the block supporting $Δ_{λ}$ corresponds to exactly the block supporting $S_{λ}$ . Therefore $Δ_{λ}, Δ_{μ}$ share a $C_{n}$ -block if and only if $S_{λ}, S_{μ}$ share an $H_{n}$ -block, and [F3] describes both sides as linkage by chains of cell modules with neighbouring terms sharing a simple composition factor; step 2.2 supplies the multiplicity transfers when a nonzero simple is needed. No equality-of-residue-contents assertion is used.

Remarks

The theorem is the algebraic half of the block theory of this page: it proves that the Schur standards and Specht standards have matching heads and matching block partitions, using the semistandard cellular basis, the explicit center isomorphism, and the corner functor. The residue-content/Jantzen computation that identifies the blocks with combinatorial data is a separate theorem with its own suppliers. No choice principle is used: all cell layers, bases, and composition series are finite.

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Generic cyclotomic Schur deformation standards have nondegenerate forms

Statement

Over the fraction field of the universal Laurent parameter ring, all cyclotomic Specht modules are simple and $H_{n}$ is split semisimple; all Schur standards $Δ_{λ}$ equal $L_{λ}$ and have nondegenerate cellular forms. Moreover, fix $N \geq 1$ , a field $F$ with nonzero $q_{H}, Q_{1}, \dots, Q_{ℓ}$ , and $0 \leq d \leq N$ , and take $O = F [t, t^{- 1}]_{(t - 1)}$ with $z = q_{H} t$ , $U_{a} = Q_{a} t^{N a}$ . Over $F (t)$ , all rank- $d$ Specht and Schur standards are simple and their forms are nondegenerate. This includes the charged finite- $e$ specialization used by the Lyle–Mathas deformation. The lemma establishes generic nondegeneracy only; it does not prove any signed-rim-hook weight-Gram determinant valuation formula.

Facts & Assumptions

Given: The cyclotomic algebra and its Specht modules [F1], the cellular Murphy basis [F2], the Jucys–Murphy content triangularity [F3], the finite cellular-head theory [F4], the semistandard Schur endomorphism basis [F5], and the Schur standard/Specht correspondence [F6].

[F1]

$H_{n}$ , its multipartition Specht (cell) modules $S_{λ}$ with cell form and quotient $D_{λ} = S_{λ} / r a d S_{λ}$ , and the Schur datum $C_{n}$ with standards $Δ_{λ}$ and heads $L_{λ} = Δ_{λ} / r a d Δ_{λ}$ (Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules).

[F2]

The $m_{s t}$ are an $R$ -basis of $H_{n}$ with the dominance-cellular multiplication law, so $H_{n}$ has a finite filtration whose successive right-module layers are copies of the right cell modules (The arbitrary-level cyclotomic Hecke Murphy basis is cellular).

[F3]

On $S_{λ}$ the elements $X_{r}$ act dominance-triangularly with diagonal $U_{a} z^{j - i}$ at the node $(i, j, a)$ carrying $r$ , and the universal triangularity specializes to the generic fibre, so $S_{λ}$ has a simultaneous triangular filtration whose successive characters are its own word characters (Arbitrary-level Murphy Jucys–Murphy action is content triangular).

[F4]

Over a field, every nonzero cellular head quotient $D_{λ}$ is absolutely simple, the distinct nonzero $D_{λ}$ exhaust the simple modules, and $[S_{λ} : D_{λ}] = 1$ with $[S_{λ} : D_{μ}] \neq 0$ only for $λ ⊵ μ$ (Nonzero-form cellular heads, cell purity and shared-factor linkage).

[F5]

The semistandard sums give the actual cellular endomorphism basis of $C_{n}$ , with standards $Δ_{λ}$ and their cellular bilinear forms defined over the universal ring; this supplier asserts no nondegenerate-or-zero dichotomy (Cyclotomic semistandard sums give the actual Schur endomorphism basis).

[F6]

The corner functor gives $Δ_{λ} e_{ω} ≅ S_{λ}$ and $L_{λ} e_{ω} ≅ D_{λ}$ (possibly $0$ ), with $[Δ_{λ} : L_{μ}] = [S_{λ} : D_{μ}]$ whenever $D_{μ} \neq 0$ , and every $Δ_{λ}$ is nonzero and block-pure (Cyclotomic Schur standards and corner functor control block linkage).

Proof

technique · direct

Given: The data and suppliers of [F1]–[F6]; let $K$ be the fraction field of the universal Laurent parameter ring.

1.1F1F3algebra

Word separation over the universal fraction field. Assign to each standard multitableau the sequence of joint Jucys–Murphy words: the diagonal values $U_{a} z^{j - i}$ at the nodes carrying the labels $1, \dots, n$ ([F3]). If two standard multitableaux first differ at step $r$ , then their restrictions to the boxes $1, \dots, r - 1$ coincide, and the two added nodes are two different addable nodes of that common prefix shape. If the two nodes lie in one physical component, their contents $j - i$ are different (two addable nodes of a partition have different contents), and $z$ is transcendental over the base field, so $z^{j - i} \neq z^{j^{'} - i^{'}}$ . If they lie in different components $a \neq b$ , the values $U_{a} z^{j - i}$ and $U_{b} z^{j^{'} - i^{'}}$ are distinct in $K$ , because $z, U_{1}, \dots, U_{ℓ}$ are algebraically independent monomials and the components differ. Hence the joint word determines the tableau, different standard multitableaux -- even of different shapes -- have different joint words, and distinct shapes have disjoint sets of word characters.

2.1F2F3F4step 1.1algebra

Generic simplicity and split semisimplicity. By [F3] each $S_{λ}$ has a simultaneous triangular filtration whose successive characters are its own words of step 1.1. Hence every composition factor $D_{μ}$ of the nonzero module $S_{λ}$ has a word occurring in $S_{μ}$ and a word occurring in $S_{λ}$ ; disjointness of shape word sets forces $μ = λ$ . In particular $D_{λ} \neq 0$ , and the cellular diagonal multiplicity $[S_{λ} : D_{λ}] = 1$ of [F4] forces $S_{λ} = D_{λ}$ to be simple, with the distinct absolutely simple modules $S_{λ}$ as the only composition factors. The cellular basis gives $\dim_{K} H_{K} = \sum_{λ} (\dim_{K} S_{λ})^{2}$ . The semisimple quotient is a product of full matrix algebras on the exhaustive absolutely simple $S_{λ}$ : to see this, the rank-one cell action on each simple gives every matrix unit, and the distinct simple labels give the distinct simple factors. Its dimension is the same sum, so the radical has dimension zero. Thus $H_{K}$ is split semisimple; no splitting of the cellular filtration was assumed.

3.1F1F3step 1.1algebra

Word separation for the Lyle–Mathas deformation. Fix $N \geq 1$ , nonzero $q_{H}, Q_{a}$ in $F$ , and set $z = q_{H} t$ , $U_{a} = Q_{a} t^{N a}$ in $F (t)$ . For two standard multitableaux that first differ at a step $r \leq N$ , the common prefix of size $r - 1$ has components of sizes summing to $r - 1$ ; the two added nodes lie in components $a, b$ and each added content $c = j - i$ is bounded in absolute value by the size of its own component, so $∣ c - c^{'} ∣ < r \leq N$ . A collision $U_{a} z^{c} = U_{b} z^{c^{'}}$ with $a \neq b$ would force the equality $N a + c = N b + c^{'}$ of $t$ -adic exponents, i.e. the nonzero multiple $N (a - b)$ of $N$ would equal $c^{'} - c$ with $∣ c^{'} - c ∣ < N$ , impossible. If $a = b$ , then $z$ is transcendental over $F$ , so $z^{c} = z^{c^{'}}$ forces $c = c^{'}$ . Thus the joint words separate all rank- $d$ standards in $F (t)$ as in step 1.1, and the argument of step 2.1 proves that the rank- $d$ Specht modules are simple and the corresponding Hecke algebra is split semisimple over $F (t)$ , with no affine-center or block-classification input.

4.1F5F6step 2.1step 3.1algebra∎

Nondegenerate Schur standards. By [F6], over any field with unit parameters, $[Δ_{λ} : L_{μ}] = [S_{λ} : D_{μ}]$ whenever $D_{μ} \neq 0$ , and every $Δ_{λ}$ is nonzero. Over $K$ , step 2.1 makes every $S_{λ}$ simple; hence $D_{μ} = 0$ never occurs, $[Δ_{λ} : L_{λ}] = [S_{λ} : S_{λ}] = 1$ , and since all $L_{μ}$ are distinct the multiplicity list forces $Δ_{λ} = L_{λ}$ to be simple; the diagonal cell-form entry of the standard of shape and type $λ$ is $1$ , so the cellular form is nonzero and, being invariant on a simple module, is nondegenerate. The same argument applies over $F (t)$ to the rank- $d$ standards using step 3.1, and the construction of [F5] and the multiplicities of [F6] are identities over the respective ground fields, so the charged finite- $e$ specializations used by the Lyle–Mathas deformation are included. No valuation of a weight-Gram determinant is computed.

Remarks

The lemma is exactly the generic-nondegeneracy input of the Jantzen theory: it upgrades the cellular form theory to simplicity of all standards over the two generic regimes, so that the Jantzen filtrations begin from nondegenerate forms. The signed rim-hook weight-Gram determinant valuations are a separate theorem with its own suppliers. No choice principle is used: all tableaux, words, and composition series are finite.

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Cyclotomic Schur weight Gram determinants are products of monic semistandard norms

Statement

Work with the arbitrary-level Murphy/Schur datum of Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules over either (i) the fraction field $K$ of the universal Laurent parameter ring, including the generic field of the Lyle–Mathas deformation, or (ii) any field $F$ with nonzero $q_{H}, Q_{1}, \dots, Q_{ℓ}$ under the deformation $z = q_{H} t$ , $U_{a} = Q_{a} t^{n a}$ in $F (t)$ . Let $b_{λ}$ be the Murphy cellular form on the Specht module $V_{λ}$ with basis $v_{s}$ over standard tableaux $s$ and $v_{0} = v_{t^{λ}}$ ; put $B_{k} = \sum_{a \leq k} ∣ ν^{(a)} ∣$ , $u_{ν} = \prod_{k = 2}^{ℓ} \prod_{r = 1}^{B_{k - 1}} (X_{r} - U_{k})$ , $x_{ν} = \sum_{w \in S_{ν}} T_{w}$ , $m_{ν} = u_{ν} x_{ν}$ , let $A_{ν}$ be the action of $m_{ν}$ on $V_{λ}$ , let $P_{ν} (z) = \sum_{w \in S_{ν}} z^{ℓ (w)}$ , and let $f_{s} = v_{s} + \sum_{r ▹ s} a_{s r} v_{r}, γ_{s} = b_{λ} (f_{s}, f_{s})$ be the monic simultaneous Jucys–Murphy eigenbasis and its norms. Let $G_{λ ν}$ be the Schur cellular Gram matrix on the type- $ν$ basis of $Δ (λ)$ , indexed by the semistandard $T$ of shape $λ$ and type $ν$ , and for such $T$ let $t_{T}$ be its dominance-minimal standard refinement, $r_{x} = U_{a (x)} z^{content (x)}$ the residue of the node $x$ of $λ$ , and $a_{T} (x)$ the type component of the label of $x$ . Then $\det G_{λ ν} = \prod_{T \in {SStd}_{ν} (λ)} \frac{γ_{t_{T}}}{C_{t_{T}} u_{T}}, C_{t} = \prod_{i < j s a m e t y p e r o w} \frac{z r_{t} (j) - r_{t} (i)}{r_{t} (j) - r_{t} (i)}, u_{T} = \prod_{x} \prod_{b > a_{T} (x)} (r_{x} - U_{b}),$ with empty determinant $1$ . The monic transition determinant from the semistandard basis $(w_{T})$ to the chain basis $(g_{T})$ is exactly $1$ . Equivalently, with $C (A, B) = \frac{z B - A}{B - A}$ , $R (A, B) = \frac{(z B - A) (B - z A)}{(B - A)^{2}}$ and $C (T), I (T)$ the pair lists of the proof, $\det G_{λ ν} = \prod_{T} B_{T} \prod_{x < y} C (r_{x}, r_{y})^{c_{x y}} R (r_{x}, r_{y})^{i_{x y}}, B_{T} = \prod_{x} \prod_{a (x) < b \leq a_{T} (x)} (r_{x} - U_{b}),$ where $c_{x y}, i_{x y}$ are explicit nonnegative semistandard counts. All factors are nonzero in the stated generic fields, and no quantum factorial is inverted in the $O$ -lattice; arbitrary composition series are auxiliary only.

Facts & Assumptions

Given: The Murphy/Schur datum [F1], the affine crossed product [F2], the cellular Murphy basis [F3], the Jucys–Murphy content triangularity [F4], the permutation-module basis [F5], the Schur cellular maps [F6], the generic nondegeneracy [F7], and the determinant rules [F8].

[F1]

$H_{n}$ , its Specht modules $V_{λ} = S_{λ}$ with cellular form, the commuting invertible generators $X_{r}$ (the Jucys–Murphy elements), the row subgroups, $m_{ν}$ , the Schur algebra $C_{n}$ with standards $Δ (λ)$ and their cellular forms (Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules).

[F2]

In the localization of the affine algebra at the variable differences the algebra maps into the rational crossed product with $T_{i} = c_{i, i + 1} + d_{i, i + 1} s_{i}$ , $c_{i j} = (z - 1) X_{j} / (X_{j} - X_{i})$ , $d_{i j} = (X_{j} - z X_{i}) / (X_{j} - X_{i})$ , and the $Φ_{w}$ over reduced $w$ are a basis there (Universal nondegenerate affine Hecke ordered monomials form a basis).

[F3]

The $m_{s t}$ are an $R$ -basis of $H_{n}$ with the dominance-cellular law, coefficients independent of the left index, and the flag condition on semistandard tableaux (The arbitrary-level cyclotomic Hecke Murphy basis is cellular).

[F4]

Distinct generic Jucys–Murphy words separate standard tableaux; $X_{r}$ acts on $S_{λ}$ dominance-triangularly with diagonal $U_{a} z^{j - i}$ at the node carrying $r$ (Arbitrary-level Murphy Jucys–Murphy action is content triangular).

[F5]

$m_{ν} H_{n}$ has the semistandard basis $m_{(T, t)} = \sum_{s : ν (s) = T} m_{s t}$ and the integral membership identity $m_{ν} h_{T} = m_{(T, t^{λ})}$ for an actual $h_{T}$ (Cyclotomic semistandard row-type sums give permutation-module bases).

[F6]

The Schur cellular maps $φ_{S T}$ and their multiplication law are established, with $Δ (λ)$ the cell module (Cyclotomic semistandard sums give the actual Schur endomorphism basis).

[F7]

Over the two generic regimes above all Specht and Schur standards are simple with nondegenerate forms (Generic cyclotomic Schur deformation standards have nondegenerate forms).

[F8]

Determinants are multiplicative, $\det X^{T} = \det X$ , and the determinant of a block-triangular matrix is the product of the diagonal block determinants (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix, For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ , For every square matrix over a commutative ring, $\det (A^{T}) = \det (A)$ ).

Proof

technique · direct

Given: The data and suppliers of [F1]–[F8]; the two generic regimes of the Statement.

1.1F1F4F5F7algebra

Generic diagonalizability of $A_{ν}$ . The finite row-coset identity gives $x_{ν}^{2} = P_{ν} (z) x_{ν}$ , and every factor of $P_{ν} (q_{H} t) = \prod_{R} \prod_{j = 1}^{∣ R ∣} (1 + q_{H} t + \dots + (q_{H} t)^{j - 1})$ is a nonzero polynomial over any $F$ , so $P_{ν} \neq 0$ in $K$ and $x_{ν} / P_{ν}$ is a selfadjoint idempotent. The polynomial $u_{ν}$ is symmetric separately on each initial component interval, hence commutes with every row generator and with $x_{ν}$ , and both are selfadjoint. Distinct generic JM words provide a basis of simultaneous eigenvectors ([F4]); on the eigenline of $s$ , the element $u_{ν}$ has eigenvalue $d_{s} = \prod_{k = 2}^{ℓ} \prod_{r = 1}^{B_{k - 1}} (r e s (s^{- 1} (r)) - U_{k})$ , so $u_{ν}$ is diagonalizable, and since $x_{ν}$ commutes with $u_{ν}$ , each $u_{ν}$ -eigenspace splits into the image and kernel of the idempotent $x_{ν} / P_{ν}$ . Hence $A_{ν} = x_{ν} u_{ν}$ is diagonalizable with nonzero eigenvalues $α = P_{ν} d_{s} \neq 0$ . Put $E = i m A_{ν}$ ; then $V_{λ} = E \oplus \ker A_{ν}$ is an orthogonal decomposition by selfadjointness, $A_{ν} ∣_{E}$ is invertible and selfadjoint, and the supplied semistandard basis [F5] identifies the $λ$ -cell layer of $m_{ν} H_{n}$ with the span of the tensors $w_{T} \otimes v_{s}$ , $w_{T} = \sum_{s : ν (s) = T} v_{s}$ , so on the generic simple block [F7] the $(w_{T})_{T}$ are a basis of $E$ and $\dim E = K_{(λ, ν)}$ .

2.1F5F6step 1.1algebra

The inverse-operator Gram formula. The cell quotient of the membership identity of [F5], followed by orthogonal projection to $E$ and pairing with $w_{T} \in E$ (which annihilates $\ker A_{ν}$ ), gives $m_{(t^{λ}, T)} ρ (h_{U}) = v_{0} b_{λ} (w_{T}, (A_{ν} ∣_{E})^{- 1} w_{U}) b_{λ} (v_{0}, -)$ . On the Schur side the composite $M^{λ} \to M^{ν} \to M^{λ}$ sending $m_{λ}$ to $m_{(t^{λ}, T)} h_{U}$ is $G_{λ ν} (T, U) m_{λ}$ modulo higher shapes by the cellular law [F6], and higher Schur maps evaluate to higher Murphy shapes and therefore vanish on this block. Comparing the resulting rank-one operators, and observing that changing $h_{U}$ changes it only by a map into $\ker A_{ν}$ , proves $G_{λ ν} (T, U) = b_{λ} (w_{T}, (A_{ν} ∣_{E})^{- 1} w_{U})$ as an identity over the ground field. This formula uses the inverse operator rather than identifying the Schur form with a restriction of the Specht form.

2.2F1F3F4step 1.1algebra

The chain basis and the unitriangular transition. Let $R_{1} < \dots < R_{m} = n$ be the ends of the consecutive nonempty rows of the type tableau of $ν$ . A semistandard $T$ determines the chain $λ (T) = (λ_{T} [1], \dots, λ_{T} [m])$ of prefix shapes filled by its first $j$ type rows; every standard refinement in the fibre of $T$ has prefix shape $λ_{T} [j]$ on its first $R_{j}$ entries, and conversely the chain determines $T$ , so chains and semistandard tableaux of type $ν$ correspond bijectively. The elementary symmetric polynomials in $X_{1}, \dots, X_{R_{j}}$ , for every row end $R_{j}$ , commute with every row generator (a type row lies wholly inside or wholly outside the prefix and Bernstein commutation fixes symmetric polynomials), hence with $m_{ν}$ , and thus preserve $E$ ; they act on the chain- $T$ layer by the elementary symmetric functions of its prefix residue multiset. Distinct multipartitions have distinct residue multisets over $K$ (distinct $U_{a}$ separate components, and within a component the content-diagonal multiplicities recover the partition), so these operators separate the chains. Ordering $T$ by a linear extension of dominance of the chains, the action on $E$ is triangular in the chain order with diagonal the chain scalar; commuting diagonalizable operators are jointly diagonalized, and the projector $E_{T}$ onto the chain- $T$ eigenspace is $1$ on the fibre of $T$ and $0$ on all other fibres. Hence $g_{T} = w_{T} E_{T} = w_{T} + \sum_{U : λ (U) > λ (T)} a_{T U} w_{U},$ the $g_{T}$ form a basis of $E$ , the transition matrix from $(w_{T})$ to $(g_{T})$ is upper unitriangular with determinant exactly $1$ , and $b_{λ} (g_{T}, g_{U}) = 0$ for $T \neq U$ because differing chains are separated by a selfadjoint row-prefix operator.

3.1F8step 2.1algebra

Spectral-block determinant. Decompose $E = ⨁_{α \neq 0} E_{α}$ into $A_{ν}$ -eigenspaces; they are mutually orthogonal with nondegenerate restricted forms. Choose bases $B_{α}$ of the $E_{α}$ , let $H_{α}$ be the $b_{λ}$ -Gram matrix of $B_{α}$ , and let $X$ be the coordinate matrix whose columns are the $w_{T}$ in the concatenated basis. Step 2.1 gives $G_{λ ν} = X^{T} {d i a g}_{α} (α^{- 1} H_{α}) X$ , hence by [F8] $\det G_{λ ν} = (\det X)^{2} \prod_{α \neq 0} α^{- \dim E_{α}} \det H_{α} .$ A change of the bases $B_{α}$ multiplies each $\det H_{α}$ by $(\det R_{α})^{2}$ and divides $\det X$ by $\prod_{α} \det R_{α}$ , so the expression is independent of the arbitrary choices; individual terms may have negative valuation while the total is integral because the Schur Gram matrix is integral. No diagonalization of a symmetric form is used, so characteristic $2$ causes no extra issue.

3.2F3step 2.2algebra

Normalization by the minimal refinement. Let $t_{T}$ be the dominance-minimal standard refinement of $T$ : in each type-row class place the next increasing interval into its nodes in reverse physical-component order, then reverse physical-row order, then left to right. The eigenvalue of $u_{ν}$ on $f_{t_{T}}$ is $u_{T} = \prod_{x} \prod_{b > a_{T} (x)} (r_{x} - U_{b})$ , which is nonzero in the stated generic fields because the flag condition [F3] places the node $x$ in a physical component strictly earlier than $b$ , so $r_{x}$ and $U_{b}$ are distinct monomials. Since $u_{ν}$ commutes with $x_{ν}$ , we have $f_{t_{T}} x_{ν} = u_{T}^{- 1} f_{t_{T}} m_{ν} \in E$ , and this vector lies in the chain- $T$ eigenspace, hence is a scalar multiple of $g_{T}$ ; the scalar is $c_{T} = [f_{t_{T}}] (f_{t_{T}} x_{ν})$ , and comparing the coefficient of $v_{t_{T}}$ — which equals $1$ in $g_{T}$ , while no strictly smaller refinement lies in that chain — gives $g_{T} = c_{T}^{- 1} f_{t_{T}} x_{ν}$ with the nonzero value of $c_{T}$ computed below.

3.3F3F4step 2.2algebra

Initial norm and adjacent-swap recurrence. For the initial tableau $f_{t^{λ}} = v_{t^{λ}}$ , and the cellular Gram entry is the coefficient of $m_{λ}$ in $m_{λ}^{2}$ modulo higher shapes, so with $P_{λ} (z) = \prod_{rows r} [∣ r ∣]_{z}!$ and the row generators acting on $v_{t^{λ}}$ by $z$ and $u_{λ}$ by its residue product, $γ_{t^{λ}} = P_{λ} (z) \prod_{a = 1}^{ℓ} \prod_{x \in [λ^{(a)}]} \prod_{b > a} (U_{a} z^{content (x)} - U_{b}) .$ If $t$ strictly dominates $s = t (i, i + 1)$ with $s$ standard and $A = r_{t} (i)$ , $B = r_{t} (i + 1)$ , then $f_{t} T_{i}$ is supported on $f_{t}, f_{s}$ , with diagonal coefficients $a_{t} = (z - 1) B / (B - A)$ , $a_{s} = (z - 1) A / (A - B)$ satisfying $a_{t} + a_{s} = z - 1$ , off-diagonal coefficient $1$ in the monic normalization, and reverse coefficient $b_{s} = z + (z - 1) a_{t} - a_{t}^{2} = (z - a_{t}) (1 + a_{t})$ ; selfadjointness and orthogonality give the recurrence $γ_{s} = γ_{t} \frac{(z B - A) (B - z A)}{(B - A)^{2}} = γ_{t} R (A, B),$ whose denominator is nonzero because a permitted standard swap concerns incomparable nodes.

4.1step 2.1step 2.2step 3.1step 3.2algebra

The monic Schur norm and the determinant product. Selfadjointness and $x_{ν}^{2} = P_{ν} x_{ν}$ give $b_{λ} (f_{t_{T}} x_{ν}, f_{t_{T}} x_{ν}) = P_{ν} b_{λ} (f_{t_{T}}, f_{t_{T}} x_{ν}) = P_{ν} c_{T} γ_{t_{T}}$ , and $A_{ν} g_{T} = P_{ν} u_{T} g_{T}$ because $A_{ν}$ acts as $P_{ν} u_{T}$ on the chain- $T$ eigenspace. Using step 2.1 in the $g$ -basis (the transition from $(w_{T})$ has determinant $1$ ) together with the orthogonality of step 2.2, $b_{λ} (g_{T}, (A_{ν} ∣_{E})^{- 1} g_{T}) = \frac{b_{λ} (g_{T}, g_{T})}{P_{ν} u_{T}} = \frac{γ_{t_{T}}}{c_{T} u_{T}},$ and the off-diagonal entries vanish, so $\det G_{λ ν} = \prod_{T \in {SStd}_{ν} (λ)} \frac{γ_{t_{T}}}{c_{T} u_{T}} .$

4.2F2F3step 3.2algebra

The explicit symmetrizer value of $c_{T}$ . We use [F2] only for the row parabolic, localized at differences of variables in the same type-row interval. This restriction is essential: differences between arbitrary JM entries need not be invertible on a Specht module. The chain- $T$ subspace of step 2.2 is stable under the row parabolic; its JM eigenwords are precisely the standard refinements of $T$ , so the entries in each type row occupy a horizontal strip in each physical component. Two nodes of a horizontal strip in one component have distinct contents: for upper row $i$ and lower row $j$ , every strip node of row $i$ lies strictly right of every strip node of row $j$ , making the contents strictly ordered. For different components the content difference is at most $n - 2$ (each node has absolute content bounded by its component size minus one), so it cannot cancel the nonzero multiple $n (a - b)$ in the deformation; in the universal field the independent $U_{a}$ separate them. Thus every same-row variable difference acts invertibly on this entire chain subspace. The finite row Bernstein relations therefore extend its action to this row-local algebra. The crossed-product coefficient calculation below is an identity in that row-local algebra and can be evaluated here, without extending the action to the localization at all variable differences. For a row interval $I$ , put $x_{I} = \sum_{w \in S_{I}} T_{w}$ , write $x_{I} = \sum_{w} a_{w} w$ ; the row-coset identity $T_{i} x_{I} = z x_{I}$ gives $a_{s_{i} w} = s_{i} (a_{w})$ by coefficient comparison with $T_{i} = c_{i, i + 1} + d_{i, i + 1} s_{i}$ , and the coefficient of the longest permutation comes from the product of the $d_{i j} = (X_{j} - z X_{i}) / (X_{j} - X_{i})$ over the positive roots of $I$ , each once. Hence $a_{1} = w_{0} (\prod_{i < j \in I} \frac{X_{j} - z X_{i}}{X_{j} - X_{i}}) = \prod_{i < j \in I} \frac{z X_{j} - X_{i}}{X_{j} - X_{i}} = C_{I} (X)$ . Disjoint row intervals commute, so $x_{ν} = \prod_{I} x_{I}$ has identity coefficient $\prod_{I} C_{I} (X)$ ; expanding $x_{I} = C_{I} (X) + \sum_{w \neq 1} b_{w} (X) Φ_{w}$ and using that every $Φ_{w}$ with $w \neq 1$ changes the distinct row entries and hence the JM word, the diagonal coefficient of $f_{t_{T}} x_{ν}$ is exactly $C_{t_{T}}$ evaluated at the residues of $t_{T}$ . Therefore $c_{T} = C_{t_{T}} = \prod_{i < j same type row} \frac{z r_{t_{T}} (j) - r_{t_{T}} (i)}{r_{t_{T}} (j) - r_{t_{T}} (i)}$ . Every factor is nonzero in the stated generic fields: in one physical component two nodes of a horizontal strip have distinct contents, and in the deformation two nodes of different components have content difference at most $n - 2$ , which cannot cancel the exponent $n (a - b)$ .

5.1F1step 4.1step 3.3step 4.2algebra∎

The inversion-set formula and the explicit product. Every standard tableau is a linear extension of the node order; bubble-sorting it toward the initial extension swaps adjacent inverted (hence incomparable) nodes exactly once and never swaps comparable ones, so reversing those swaps and multiplying the recurrence of step 3.3 gives $γ_{t_{T}} = γ_{t^{λ}} \prod_{\begin{array}{c} x < y \\ t_{T} (x) > t_{T} (y) \end{array}} R (r_{x}, r_{y}),$ and $R$ is symmetric in its arguments. Dividing by $C_{t_{T}}$ and $u_{T}$ and using $γ_{t^{λ}} / u_{T} = P_{λ} (z) B_{T}$ with $B_{T} = \prod_{x} \prod_{a (x) < b \leq a_{T} (x)} (r_{x} - U_{b})$ (a literal nested-product cancellation by the flag condition), together with the telescoping row identity $\prod_{i < j} C (A_{i}, A_{j}) = [r]_{z}!$ for residues $A_{j} = A_{1} z^{j - 1}$ in a row of length $r$ , the four pair classes — same physical row with same or different labels, different physical rows with same or different labels — give exactly $\frac{γ_{t_{T}}}{C_{t_{T}} u_{T}} = B_{T} \prod_{(x, y) \in C (T)} C (r_{x}, r_{y}) \prod_{(x, y) \in I (T)} R (r_{x}, r_{y}),$ where $C (T)$ collects pairs $x < y$ that are in one physical row with different labels or in different physical rows with the same label, and $I (T)$ collects pairs $x < y$ in different physical rows with $L_{T} (x) > L_{T} (y)$ . Multiplying over $T$ and collecting like node pairs yields $\det G_{λ ν} = \prod_{T} B_{T} \prod_{x < y} C (r_{x}, r_{y})^{c_{x y}} R (r_{x}, r_{y})^{i_{x y}}, c_{x y} = # {T : (x, y) \in C (T)}, i_{x y} = # {T : (x, y) \in I (T)} .$ All factors are nonzero in the two generic regimes by step 4.2, the empty weight contributes determinant $1$ , and no factor $P_{ν}$ or $c_{T}$ is inverted in the $O$ -lattice; this proves both displayed determinant formulas.

Remarks

The lemma is the determinant input of the Jantzen sum formula: it replaces the signed rim-hook valuation by an exact product of monic semistandard norms, with the transition determinant computed rather than bounded. The inverse operator $A_{ν} ∣_{E}^{- 1}$ is essential; the Schur form is not the restriction of the Specht form except at the regular weight. No choice principle is used: all tableaux, chains, and matrices are finite.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Cyclotomic Schur weight Gram valuations equal signed rim-hook Kostka sums

Statement

Fix any field $F$ , level $ℓ \geq 1$ , $q_{H} \in F^{\times}$ of finite order $e \geq 2$ , charges $Q_{a} = q_{H}^{c_{a}}$ , $n \geq 0$ , and the Lyle–Mathas parameters $O = F [t, t^{- 1}]_{(t - 1)}$ , $z = q_{H} t$ , $U_{a} = Q_{a} t^{n a}$ , with $V = ν_{t - 1} \circ (-)$ . Then for every multipartition shape $λ$ and actual multipartition Schur weight $ν$ , $V (\det G_{λ ν}) = \sum_{μ} J_{λ μ} K_{μ ν},$ where $K_{μ ν}$ is the semistandard Kostka number of Semistandard tableaux and Kostka numbers and $J_{λ μ} = 0$ unless $λ$ strictly dominates $μ$ ; otherwise $J_{λ μ}$ is the sum over all pairs of equal-length rim hooks removed from $λ$ and from $μ$ to the same remainder of $(- 1)^{{leg}_{λ} + {leg}_{μ}}$ times $V ({r e s}_{O} ({foot}_{λ}) - {r e s}_{O} ({foot}_{μ}))$ , the modulus $ν_{t - 1}$ of the Laurent difference of the two foot residues. Every such difference is nonzero in $F (t)$ . The full signed marked-strip commutator and its auxiliary coefficient identity are proved locally below; arbitrary-composition coefficients are combinatorial only and never introduce extra Schur modules.

Facts & Assumptions

Given: The monic weight-Gram supplier [F1], the Laurent-binomial valuation leaf [F2], the multipartition and Schur conventions [F3], the tableau conventions [F4]–[F5], and the determinant convention [F6].

[F1]

The exact monic product $\det G_{λ ν} = \prod_{T} γ_{t_{T}} / (C_{t_{T}} u_{T})$ with the explicit node-pair factors $C_{+} (A, B) = (z B - A) / (B - A)$ and $R (A, B) = (z B - A) (B - z A) / (B - A)^{2}$ (Cyclotomic Schur weight Gram determinants are products of monic semistandard norms).

[F2]

For $k > 0$ and $z = q_{H} t$ , $V (z^{k} - 1) = 0$ if $q_{H}^{k} \neq 1$ . If $q_{H}^{k} = 1$ , it is $1$ in characteristic zero and $p^{v_{p} (k)}$ in characteristic $p$ , and $V$ is a valuation with $V (a b) = V (a) + V (b)$ ; the valuation of a nonzero constant is $0$ (Laurent binomial valuation at one over any field).

[F3]

Multipartitions, their dominance order, the flag condition and the multipartition parameters $z, U_{a}$ over $O$ (Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules).

[F4]

Standard tableaux, shapes, and contents (Tableaux and standard tableaux).

[F5]

Semistandard tableaux, weight, and the Kostka number (Semistandard tableaux and Kostka numbers).

[F6]

Determinants of square matrices over a commutative ring (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

Proof

technique · direct

Given: The data and suppliers of [F1]–[F6]; work throughout with the fixed final-rank- $n$ parameters $z = q_{H} t$ , $U_{a} = Q_{a} t^{n a}$ at every auxiliary rank.

1.1F3F4F5algebra

Characters as finite symmetric polynomials. Fix $d \geq n$ row labels per type component with ordered variables $x_{(a, r)}$ , put $Y_{a} = {x_{(b, r)} : b \geq a}$ with the inherited order, and for a shape $λ$ let ${ch}_{λ} (x) = \sum_{T \in SStd (λ)} x^{wt (T)}$ . The flag condition says that the physical component $a$ of a semistandard tableau is an ordinary semistandard tableau over the alphabet $Y_{a}$ , and the components are chosen independently, so ${ch}_{λ} (x) = \prod_{a} s_{λ^{(a)}} (Y_{a})$ with $s_{ρ} (Y) = \det (h_{ρ_{i} - i + j} (Y))$ . The determinant equals the ordinary semistandard generating polynomial: expand it as signed families of east/north lattice paths from $(- j, 1)$ to $(ρ_{i} - i, m)$ with east steps at height $r$ weighted $y_{r}$ , choose the first intersection in a fixed order and exchange the two tails, an involution that preserves the monomial weight and reverses the permutation sign, so all intersecting families cancel; a nonintersecting family forces the identity matching, and the surviving families are exactly the semistandard tableaux, sign $+ 1$ . Hence the coefficient of $x^{ν}$ in ${ch}_{λ}$ is the Kostka number $K_{λ ν}$ ; empty and short-row cases are included.

2.1F4step 1.1algebra

Beta determinants and rim-hook signs. Choose $N \geq n$ , put $β_{a, i} = λ_{i}^{(a)} + N - i$ with zero parts appended, and for a nonnegative integer array $v$ define $χ (v; x) = \prod_{a} \det (h_{v_{a, i} - N + j} (Y_{a}))_{i, j}$ . If two entries of one component of $v$ agree the determinant has equal rows and is zero; otherwise sorting that component into strictly decreasing order and writing $ρ_{i} = v_{i}^{'} - N + i$ gives $χ (v; x) = ε (v) \prod_{a} s_{ρ^{(a)}} (Y_{a})$ with $ε$ the product of the row-sorting signs, so every coefficient is a signed Kostka multiplicity. A beta move lowering a bead from $v + L$ to an unoccupied $v$ is exactly the removal of a rim hook of length $L$ : tracing the southeast boundary shows the strip between the two boundary steps has $L$ boxes, its leg is the number of occupied positions strictly between $v$ and $v + L$ , its foot content is $v - N + 1$ , and the associated sorting sign is $(- 1)^{leg}$ ; the correspondence is a bijection by the reverse move.

3.1step 1.1step 2.1algebra

The hook side as a complete finite alternating polynomial. Define $B_{λ} (x)$ as the finite alternating sum of step 2.1: for crosscomponent terms, over $a < b$ , over beads $i, j$ and lengths $h \leq β_{a, i}$ , of $χ (β - h e_{a, i} + h e_{b, j}; x) V (U_{a} z^{β_{a, i} - h} - U_{b} z^{β_{b, j}})$ ; and for same-component terms, over $a$ , $i < j$ and $1 \leq h \leq β_{a, i} - β_{a, j} - 1$ , of $χ (β - h e_{a, i} + h e_{a, j}; x) V (U_{a} z^{β_{a, i} - h} - U_{a} z^{β_{a, j}})$ ; terms whose $χ$ is the zero polynomial are omitted before any valuation is evaluated. Then $B_{λ} (x) = \sum_{μ : λ ▹ μ} J_{λ μ} {ch}_{μ} (x)$ : a crosscomponent move removing from the earlier component $a$ and adding to a later component $b > a$ produces a strictly dominated $μ$ and occurs exactly once by the bead description; occupied targets give repeated rows and zero; sorting contributes $(- 1)^{removed leg + added leg}$ ; and the prefactor $z^{- N + 1}$ converts the beta difference to the difference of the deformed foot residues. In one component the two moved beads $i < j$ give a strict dominance decrease precisely for $1 \leq h < β_{i} - β_{j}$ , the exchange case $h = β_{i} - β_{j}$ giving $μ = λ$ and the outward case giving the reverse dominance; occupied intermediate positions again give zero rows, and the range exceeding a component's size gives a repeated or negative bead, hence a zero determinant. Thus no hook pair is lost or added, and the surviving summands are exactly the $J_{λ μ}$ terms.

3.2F2step 2.1algebra

Nonvanishing of every retained foot difference. For a removable rim hook of length $L$ from a partition of size $M + L$ with foot $(h, j)$ , one has $- (M + L - 1) \leq j - h \leq M$ : every box left of the foot in its row survives, so $j - 1 \leq M$ , and $h \leq M + L$ gives the lower bound. If $λ$ and $μ$ remove equal-length hooks in distinct components $a \neq b$ to the same remainder with remainder sizes $M_{a}, M_{b}$ , their foot contents satisfy $∣ γ_{x} - γ_{y} ∣ \leq M_{a} + M_{b} + L - 1 \leq n - 1$ , so the exponent $n (a - b) + γ_{x} - γ_{y}$ is a nonzero integer; in the same component equal foot content and equal length reconstruct the same inverse hook addition, hence would force $λ = μ$ , which the strict-dominance terms exclude. Therefore every retained Laurent difference $U_{a} z^{u} - U_{b} z^{v}$ is nonzero in $F (t)$ : it is a unit times a nonzero Laurent monomial times $(z^{N} - 1)$ -type factors and, by [F2], its valuation is finite, being $0$ when the constant residues differ, $1$ in characteristic zero when they agree, and $p^{v_{p} (N)}$ in characteristic $p$ when they agree.

3.3step 2.1algebra

The same-component operator. In the same-component beta sum put $D = β_{i} - β_{j}$ and $k = D - h$ , so the valuation is $f (k) = V (z^{k} - 1)$ up to unit factors; swapping the two changed determinant rows shows $χ (β - (D - k) e_{i} + (D - k) e_{j}; x) = - χ (β - k e_{i} + k e_{j}; x)$ . Hence the complete coefficient of $f (k)$ is $- \sum_{i < j : β_{i} - β_{j} > k} χ (β - k e_{i} + k e_{j}; x)$ , which is the strictly dominance-lowering part of $- p_{k} p_{k}^{⊥}$ on the abstract shape basis, where $p_{k}^{⊥}$ lowers an occupied bead by $k$ into an unoccupied nonnegative position with sign $(- 1)^{# crossed beads}$ and $p_{k}$ raises a bead by $k$ with the same sign rule; the bead-raising formula follows from the alternant identity $\det (y_{r}^{β_{i}}) \sum_{r} y_{r}^{k} = \sum_{i} \det (y_{r}^{β_{i} + k δ_{i j}})$ divided by the Vandermonde, and the removal map is its transpose. After one removal and one addition a non-diagonal lower-dominance result comes only from distinct beads $i < j$ with $0 < k < β_{i} - β_{j}$ , occupied intermediate positions giving zero repeated rows and the diagonal being removed by the strict projection; this proves the operator identification integrally, in every characteristic.

4.1step 3.1step 3.2step 3.3algebra

The crosscomponent operator as exterior currents. On the $N$ -fold exterior power of a space with basis $e_{0}, e_{1}, \dots$ , represent a partition by the ordered wedge of its beta beads and let $E (p, q)$ insert $e_{p}$ after removing $e_{q}$ , with exterior signs; it vanishes if $q$ is absent or $p$ is occupied, and its sign is the bead-sorting sign. Expanding $E = creation \cdot contraction$ gives $[E (p, q), E (r, s)] = δ_{q, r} E (p, s) - δ_{p, s} E (r, q)$ . The crosscomponent matrix acting on column shape vectors is $J^{c r o s s} = \sum_{a < b} \sum_{h > 0} \sum_{q \geq h, w \geq 0} V (U_{a} z^{q - h} - U_{b} z^{w}) E_{a} (q, q - h) E_{b} (w, w + h)$ , a finite operator on each size- $n$ sector; the reverse hook pair has forward current $E_{a} (q - h, q) E_{b} (w + h, w)$ , and transposing gives exactly this operator with the same signs and valuation.

5.1step 4.1algebra

Horizontal strips and their conjugation. With $S e_{q} = e_{q + 1}$ and $H (x) = (I - x S)^{- 1} = \sum_{r \geq 0} x^{r} S^{r}$ , whose entry $H_{p, q}$ is $x^{p - q}$ for $p \geq q$ and $0$ otherwise, the matrix of $\land^{N} H (x)$ between $β_{μ}$ and $β_{ρ}$ is $x^{∣ μ ∣ - ∣ ρ ∣} \det (1_{β_{μ, i} \geq β_{ρ, j}})$ . Subtracting consecutive columns of the $0 / 1$ matrix makes its determinant $1$ exactly when $β_{μ, i} \geq β_{ρ, i} > β_{μ, i + 1}$ for all $i$ , i.e. $μ_{i} \geq ρ_{i} \geq μ_{i + 1}$ , the horizontal-strip criterion, and $0$ otherwise; so $Γ (x) = \land^{N} H (x)$ adds every horizontal strip once with weight $x^{strip size}$ . For a type component $a$ , $Γ_{a} (x) = \prod_{b \leq a} Γ_{b} (x)$ adds a horizontal strip in every permitted physical component $b \leq a$ and leaves the rest unchanged, so the successive $Γ_{a}$ from the empty shape enumerate exactly the flagged semistandard tableaux and give the branching matrix. Conjugation is explicit: since $H^{- 1} = I - x S$ has column vector $e_{p} - x e_{p + 1}$ and row covector $\sum_{r \leq q} x^{q - r} e_{r}^{*}$ , $Γ (x)^{- 1} E (p, q) Γ (x) = \sum_{r = 0}^{q} x^{q - r} (E (p, r) - x E (p + 1, r)),$ and the two-current conjugation is the ordered product of two such sums.

6.1F1F6step 3.1step 5.1algebra

Strip factors and exact minor expressions. For a permitted strip transition $ρ \to λ$ for the next type component $c$ , let $η = [λ] ∖ [ρ]$ , $r_{x} = U_{comp (x)} z^{content (x)}$ , and put $F_{λ ρ}^{(c)} = \prod_{x \in η} \prod_{comp (x) < b \leq c} (r_{x} - U_{b}) \prod_{x \in [ρ], y \in η in one physical row} C_{+} (r_{x}, r_{y}) \prod_{x < y, x, y \in η in different physical rows} C_{+} (r_{x}, r_{y}) \prod_{x \in η, y \in [ρ], x < y} R (r_{x}, r_{y})$ . Splitting the pair lists of [F1] into old/old, old/new and new/new gives this factor, and their product along a semistandard chain is $γ_{t_{T}} / (C_{t_{T}} u_{T})$ . Choose $N > n$ , write $q_{i} = β_{λ, i}$ , $p_{i} = β_{ρ, i}$ in each component, and $H (q, p)_{i j} = 1_{q_{i} \geq p_{j}}$ , so $q_{N} = p_{N} = 0$ . Write $ϕ (k) = V (z^{k} - 1)$ for $k > 0$ and $ψ_{a b} (d) = V (U_{a} z^{d} - U_{b})$ . First regard these as independent formal symbols; canceled symbols are never evaluated. By steps 3.1 and 5.1, the same-component row contribution to $J Γ$ is $\sum_{i < j} \sum_{0 < h < q_{i} - q_{j}} ϕ (q_{i} - q_{j} - h) \det H (q - h e_{i} + h e_{j}, p)$ ; the column contribution to $Γ J$ is $\sum_{i < j} \sum_{h > 0} (ϕ (p_{i} - p_{j} + h) - ϕ (h)) \det H (q, p + h e_{i} - h e_{j})$ . Negative or repeated shifted beads give zero determinants. For the second formula, the outward source has beads $p_{i} + h, p_{j} - h$ ; its two hook pairs returning to $p$ have lengths $h$ and $p_{i} - p_{j} + h$ , foot gaps $p_{i} - p_{j} + h$ and $h$ , and opposite signs because the second pair exchanges the two target beads. This proves the difference of potentials, including crossed inverse sources. A crosscomponent row term replaces $q_{a, i}, q_{b, j}$ by $q_{a, i} - h, q_{b, j} + h$ with potential $ψ_{a b} (q_{a, i} - h - q_{b, j})$ ; its column term replaces $p_{a, i}, p_{b, j}$ by $p_{a, i} + h, p_{b, j} - h$ with potential $ψ_{a b} (p_{a, i} - p_{b, j} + h)$ . The determinants in the original indexed positions retain all sorting signs.

7.1F1step 6.1algebra

Norm endpoints. On support $q_{i} \geq p_{i} > q_{i + 1}$ . Put $M_{i} = q_{i} - p_{i}$ , $K_{j} = p_{j} - q_{j + 1} - 1$ for $j < N$ , $K_{N} = 0$ , and $S (m) = \sum_{k = 1}^{m} ϕ (k)$ . The same-component norm valuation is $\sum_{i} (S (q_{i}) - S (p_{i}) - S (M_{i})) + \sum_{i < j} \sum_{u = p_{i}}^{q_{i} - 1} (ϕ (u - q_{j}) - ϕ (u - p_{j} + 1))$ . To verify this directly, put $b_{i} = N - i$ : same-row old/new pairs sum to $S (q_{i} - b_{i}) - S (M_{i}) - S (p_{i} - b_{i})$ . For $i < j$ , new/new and new-upper/old-lower pairs sum to $\sum_{u = p_{i}}^{q_{i} - 1} (ϕ (u - q_{j}) - ϕ (u - p_{j} + 1) + ϕ (u - b_{j} + 1) - ϕ (u - b_{j}))$ . The last two terms telescope in $u$ to $ϕ (q_{i} - b_{j}) - ϕ (p_{i} - b_{j})$ ; summing over $j > i$ cancels the baseline $b_{i}$ terms and gives the formula. Every displayed $ϕ$ argument is positive. For $a < b \leq c$ , the crosscomponent endpoint expression is $\sum_{i} \sum_{u = p_{a, i}}^{q_{a, i} - 1} (ψ_{a b} (u + 1) + \sum_{j = 1}^{N} (ψ_{a b} (u - q_{b, j}) - ψ_{a b} (u - p_{b, j} + 1)))$ . Indeed the new/new $C_{+}$ factors telescope in the lower-component node index $v$ to $ψ (u - q_{b, j}) - ψ (u - p_{b, j})$ , and the new/old $R$ factors add $ψ (u - p_{b, j}) - ψ (u - b_{j})$ and the rectangle difference $\sum_{v = b_{j}}^{p_{b, j} - 1} (ψ (u + 1 - v) - ψ (u - v))$ . Telescoping this rectangle in the other direction and adding the boundary factor $ψ (u - N + 1)$ gives the displayed formula; the vacuum differences telescope and cancel that boundary. This padded formula is an equality of formal symbols: its $j = N$ term cancels $ψ (u + 1)$ before evaluation.

8.1step 6.1step 7.1algebra

All on-support minors. Here $H$ is the unit upper triangular step matrix. Expanding a replacement row or column in its basis gives $d_{i}^{r o w} (v) = 1_{p_{i} \leq v < p_{i - 1}}$ and $d_{i}^{c o l} (v) = 1_{q_{i + 1} < v \leq q_{i}}$ , with $p_{0} = \infty$ and no positive lowering from $p_{N} = 0$ . A two-row minor is its direct product minus its crossed product. For the same-component row expression, the direct term has $1 \leq h \leq \min (M_{i}, K_{j - 1})$ and the crossed term, with $k = q_{i} - q_{j} - h$ , has $1 \leq k \leq \min (M_{i}, K_{j - 1})$ . The outward two-column minor has no crossed term and its direct bound is $1 \leq h \leq \min (M_{i}, K_{j})$ . Pair direct row index $j + 1$ with first-potential column index $j$ . For arbitrary symbols $ψ$ the interval identity $\sum_{s = 0}^{M - 1} (ψ (A + s + K + 1) - ψ (A + s + 1)) = \sum_{h = 1}^{\min (M, K)} (ψ (A + M + K + 1 - h) - ψ (A + h))$ follows by canceling the overlap of the two integer intervals (when $M \leq K$ they are disjoint, and when $M > K$ only $K$ endpoints remain). Applying it with $A = p_{i} - p_{j}$ , the paired contribution is $\sum_{j = i + 1}^{N - 1} \sum_{u = p_{i}}^{q_{i} - 1} (ϕ (u - q_{j + 1}) - ϕ (u - p_{j} + 1))$ . Crossed rows cancel the second column potentials except $- S (\min (M_{i}, K_{i}))$ ; the unpaired direct row $j = i + 1$ , combined with that residual by the same identity at $A = 0$ , gives $\sum_{u = p_{i}}^{q_{i} - 1} ϕ (u - q_{i + 1}) - S (M_{i})$ . Summing yields exactly the same-component endpoints of step 7.1. For a crosscomponent pair, row $j + 1$ and column $j$ share the bound $\min (M_{i}, K_{b, j})$ and $A = p_{a, i} - p_{b, j}$ . The unpaired first row contributes $\sum_{u = p_{a, i}}^{q_{a, i} - 1} ψ (u - q_{b, 1})$ ; the final column vanishes. The same interval identity converts the paired terms to the remaining endpoints of step 7.1. Thus $(J Γ - Γ J)_{λ ρ} = x^{∣ λ ∣ - ∣ ρ ∣} V (F_{λ ρ}^{(c)})$ on support.

9.1step 6.1step 8.1algebra

Crosscomponent off support. The row of $H$ at bead $v$ is the tail vector starting at threshold $κ (v)$ , whose interval is $[p_{r}, p_{r - 1})$ . Different nonzero tail vectors are independent; hence rank equals the number of distinct thresholds. At rank $N - 1$ there is one missing threshold $r$ and two adjacent duplicated rows $i, i + 1$ . Put $Q = q_{i} > Q^{'} = q_{i + 1}$ , $[P, P^{'}) = [p_{r}, p_{r - 1})$ , $u = e_{i} - e_{i + 1}$ and $v = e_{r} - e_{r - 1}$ , omitting $e_{0}$ . The adjugate is $ε v u^{T}$ : its nullspaces have those generators, and removing a duplicate row and the appropriate column gives a minor $\pm 1$ . Thus replacement determinants are $ε u_{l} 1_{P \leq x < P^{'}}$ and $ε v_{l} 1_{Q^{'} < x \leq Q}$ with the same sign. Rank $\leq N - 2$ kills every one-row/column replacement. A defective earlier component can be repaired only when $Q^{'} \geq P^{'}$ (missing below), and a defective later component only when $Q < P$ (missing above); opposite directions give zero. If only the earlier component is defective, pair normal later row $j + 1$ and column $j$ and write $L = q_{b, j + 1}$ , $M = p_{b, j}$ . Their coefficients both give $- ε \sum_{d \in [P - L, P^{'} - L) \cap (Q^{'} - M, Q - M]} ψ (d)$ ; the first unpaired row vanishes because its two source signs sum to zero. If only the later component is defective, write its endpoints $R > R^{'}$ , $[S, S^{'})$ and let $M = q_{a, i} - p_{a, i}$ . The row coefficient at restored position $w$ is $ε 1_{R^{'} + M < w \leq R + M}$ on $[S, S^{'})$ , and the column coefficient at $t$ is $ε 1_{S \leq t + M < S^{'}}$ on $(R^{'}, R]$ . The translation $w = t + M$ matches their potentials because $q_{a, i} = p_{a, i} + M$ . With both components defective, the row coefficient of $ψ (d)$ , divided by $ε_{a} ε_{b}$ , is $(1_{S \leq Q - d < S^{'}} - 1_{S \leq Q^{'} - d < S^{'}}) (1_{P \leq R + d < P^{'}} - 1_{P \leq R^{'} + d < P^{'}})$ ; the column coefficient is $(1_{R^{'} < P - d \leq R} - 1_{R^{'} < P^{'} - d \leq R}) (1_{Q^{'} < S + d \leq Q} - 1_{Q^{'} < S^{'} + d \leq Q})$ . They agree by the endpoint identity $1_{B_{0} \leq A_{1} - d < B_{1}} - 1_{B_{0} \leq A_{0} - d < B_{1}} = 1_{A_{0} < B_{0} + d \leq A_{1}} - 1_{A_{0} < B_{1} + d \leq A_{1}}$ , obtained by regrouping four half-line indicators. Infinite top endpoints simply omit the virtual source. An untouched defective component kills the product. Components beyond $c$ are empty in the reachable prefix: the forward transfer would create a nonempty forbidden intermediate component, and the inverse transfer would remove a hook from an empty component, so both vanish. This exhausts crosscomponent off-support cases.

10.1step 6.1step 9.1algebra

Same-component deficiency one, missing below. Keep the rank-one data of step 9.1, with duplicated rows $s, s + 1$ and missing threshold $r > s$ , so $Q^{'} \geq P^{'}$ . A nonzero inward two-row replacement must use one duplicate source and a unique normal row $j \geq r + 1$ : one new row fills the missing interval and the other restores the unique source threshold. The direct and exchanged assignments have signs $ε u_{l}$ and $- ε u_{l}$ ; any other assignment leaves an unchanged duplicate or missing threshold, or violates the strict inward range. Nonzero outward two-column repairs use one endpoint $P, P^{'}$ and a unique column $j \geq r + 1$ , or the endpoint pair itself. Put $K_{j} = p_{j} - q_{j + 1} - 1$ for $j \geq r$ , $K_{N} = 0$ . The direct row contribution is $R_{j} = - ε \sum_{d \in [P - q_{j}, P^{'} - q_{j}) \cap (Q^{'} - p_{j - 1}, Q - p_{j - 1}]} ϕ (d)$ . Split each unique column contribution into its first and second potentials $C_{j}^{f i r s t} - C_{j}^{s e c o n d}$ from step 6.1. Then $R_{j + 1} = C_{j}^{f i r s t}$ , by the same interval intersection, while $C_{j}^{s e c o n d} = ε \sum_{h = 1}^{K_{j}} ϕ (h) (1_{Q^{'} < P + h \leq Q} - 1_{Q^{'} < P^{'} + h \leq Q})$ . The exchanged row is $X_{j} = - ε \sum_{k = 1}^{K_{j - 1}} ϕ (k) (1_{P \leq Q - k < P^{'}} - 1_{P \leq Q^{'} - k < P^{'}}) = - C_{j - 1}^{s e c o n d}$ by endpoint reciprocity. All terms cancel except $R_{r + 1} - C_{r}^{s e c o n d} - C^{e n d}$ . Put $A = Q^{'} - P^{'} \geq 0$ , $B = P^{'} - P > 0$ , $L = Q - Q^{'} > 0$ , $I_{0} = (A, A + L]$ , $I_{1} = I_{0} + B$ , $K = K_{r}$ . These three terms are respectively $- ε \sum_{d \in I_{1} \cap (K, K + B]} ϕ (d)$ , $ε \sum_{h = 1}^{K} ϕ (h) (1_{I_{1}} (h) - 1_{I_{0}} (h))$ , and $- ε \sum_{h \in I_{0}, 1 \leq h \leq K} (ϕ (B + h) - ϕ (h))$ ; the endpoint-pair sign is negative because the raised endpoint is $P^{'}$ . Their residual is $- ε \sum_{d \in I_{1} \cap [1, K + B]} ϕ (d) + ε \sum_{h \in I_{0}, 1 \leq h \leq K} ϕ (B + h) = 0$ by $d = h + B$ . This proves every coefficient cancels, including $K = 0$ .

10.2step 6.1step 9.1algebra

Same-component deficiency one, missing above. Here $r \leq s$ , $P > Q$ , and $ε = (- 1)^{s - r}$ , obtained by moving the inserted missing threshold from row $s$ to position $r$ ; the column cofactor has the same sign. If $r = 1$ , all inward row positions are at most $q_{1} < p_{1}$ and cannot fill the missing top threshold; the first column is zero and any outward repair raises it further, so both sums vanish. For $r \geq 2$ the only unique row sources are $i \leq r - 1$ , with $M_{i} = q_{i} - p_{i} \geq 0$ . The direct rows have $R_{i} = ε \sum_{d \in (q_{i} - P^{'}, q_{i} - P] \cap [p_{i} - Q, p_{i} - Q^{'})} ϕ (d)$ . The first column potentials equal $R_{i}$ for $i \leq r - 2$ , while $C_{i}^{s e c o n d} = ε \sum_{h = 1}^{M_{i}} ϕ (h) (1_{Q^{'} < P - h \leq Q} - 1_{Q^{'} < P^{'} - h \leq Q})$ . Exchanged rows are $X_{i} = - ε \sum_{k = 1}^{M_{i}} ϕ (k) (1_{P \leq Q + k < P^{'}} - 1_{P \leq Q^{'} + k < P^{'}}) = - C_{i}^{s e c o n d}$ , again by endpoint reciprocity. This exhausts repairs: the duplicate source must be paired with a head row above the missing interval, and outward column repairs must lower an endpoint paired with a unique head column, or use the endpoint pair. The sole residual is at $i = r - 1$ . Put $M = q_{r - 1} - P^{'} \geq 0$ , $B = P^{'} - P > 0$ , $A = P - Q > 0$ , $L = Q - Q^{'} > 0$ , $I_{0} = [A, A + L)$ , $I_{1} = I_{0} + B$ . Then $R_{r - 1} = ε \sum_{d \in I_{1} \cap (M, M + B]} ϕ (d)$ and $C_{r - 1}^{s e c o n d} = ε \sum_{h = 1}^{M} ϕ (h) (1_{I_{0}} (h) - 1_{I_{1}} (h))$ ; the endpoint-pair column term is $C^{e n d} = ε \sum_{h \in I_{0}, 1 \leq h \leq M} (ϕ (B + h) - ϕ (h))$ , with positive cofactor because the lowered endpoint is $P$ . Thus $R_{r - 1} - C_{r - 1}^{s e c o n d} - C^{e n d} = ε \sum_{d \in I_{1} \cap [1, M + B]} ϕ (d) - ε \sum_{h \in I_{0}, 1 \leq h \leq M} ϕ (B + h) = 0$ . The translation $d = h + B$ is valid since $A > 0$ .

11.1step 6.1step 9.1step 10.1step 10.2algebra

Same-component deficiency two and higher. At rank $N - 2$ , the excess threshold copies are one triple or two pairs. Inward row repairs of a triple cannot reach missing thresholds between its source beads. For two pairs with thresholds $r < s$ , both missing thresholds must lie strictly between them, say $r < m_{1} < m_{2} < s$ ; otherwise both row and column sums are zero. For columns this follows from $ℓ (p_{j}) = # {i : κ (q_{i}) \leq j}$ : duplicated row thresholds are jumps, missing thresholds are flats, and outward column repairs require two separate size-two jumps with both flats between. A zero first column also cannot be repaired by raising it. In the active pattern put $Q = q_{r} > Q^{'} = q_{r + 1}$ , $R = q_{s - 1} > R^{'} = q_{s}$ , $I_{1} = [P, P^{'}) = [p_{m_{1}}, p_{m_{1} - 1})$ , $I_{2} = [S, S^{'}) = [p_{m_{2}}, p_{m_{2} - 1})$ , $J_{Q} = (Q^{'}, Q]$ , $J_{R} = (R^{'}, R]$ . The common row/column sign is $ε = (- 1)^{m_{1} - r + s - 1 - m_{2}}$ , counted by moving the inserted thresholds to their sorted positions. Switching a source copy or switching the two target thresholds reverses it. Define $A_{I} (k) = 1_{R + k \in I} - 1_{R^{'} + k \in I}$ and $B_{I} (k) = 1_{Q - k \in I} - 1_{Q^{'} - k \in I}$ ; define $D_{P}^{\pm} (J; k) = 1_{P \pm k \in J} - 1_{P^{'} \pm k \in J}$ and $D_{S}^{\pm} (J; k) = 1_{S \pm k \in J} - 1_{S^{'} \pm k \in J}$ . The complete row coefficient of $ϕ (k)$ is $ε (A_{I_{1}} (k) B_{I_{2}} (k) - A_{I_{2}} (k) B_{I_{1}} (k))$ ; the column coefficient is $ε (D_{S}^{+} (J_{Q}; k) D_{P}^{-} (J_{R}; k) - D_{P}^{+} (J_{Q}; k) D_{S}^{-} (J_{R}; k))$ . Indeed putting $k = q_{i} - q_{j} - h$ gives row targets $q_{j} + k, q_{i} - k$ , while the positive and negative column potentials give targets $p_{j} + k, p_{i} - k$ and $p_{i} + k, p_{j} - k$ respectively. Interval membership implies all required positive-length bounds. Endpoint reciprocity gives $A_{I_{1}} = D_{P}^{-} (J_{R})$ , $A_{I_{2}} = D_{S}^{-} (J_{R})$ , $B_{I_{1}} = D_{P}^{+} (J_{Q})$ , $B_{I_{2}} = D_{S}^{+} (J_{Q})$ , proving equality. If $m_{2} = m_{1} + 1$ , the repeated column groups share an endpoint; its formal double-source term is zero because its potential is $ϕ (h) - ϕ (h)$ , so the formula still includes adjacent missing thresholds. At rank $\leq N - 3$ , two replacements cannot restore full rank. Together with steps 9.1–10.2 this proves off-support vanishing for every rank and orientation. All identities are formal coefficient equalities; retained terms are actual valid hook currents, so step 3.2 permits their finite arbitrary-characteristic valuation specialization.

12.1F1step 3.1step 3.2step 8.1step 11.1algebra∎

Telescoping and the valuation formula. Multiply the $Γ$ matrices in increasing type-row-label order and telescope $J Γ - Γ J$ from $J_{0} = 0$ ; by steps 8.1–11.1 each commutator inserts exactly $V (F)$ at its strip position, so the $λ$ -coordinate of the resulting identity is $\sum_{T} V (γ_{t_{T}} / (C_{t_{T}} u_{T})) x^{wt (T)} = L_{λ} (x)$ on one side and $B_{λ} (x)$ on the other. All padded artificial symbols cancel before the surviving coefficients are evaluated, and every surviving coefficient corresponds to a valid current, hence by step 3.2 to a nonzero foot difference with finite valuation. Comparing with the exact monic product [F1] and the hook side of step 3.1, and taking the coefficient of $x^{ν}$ at an actual multipartition weight $ν$ (auxiliary compositions only index the same polynomials and never enter the Schur conclusion), gives $V (\det G_{λ ν}) = \sum_{μ} J_{λ μ} K_{μ ν};$ the empty case $n = 0$ gives $0$ , all ranks keep the fixed parameters $z = q_{H} t$ , $U_{a} = Q_{a} t^{n a}$ , and arbitrary-composition coefficients are combinatorial only.

Remarks

The lemma converts the exact monic determinant product of the sibling into the signed rim-hook Kostka sum by an explicit exterior-current commutator and a finite marked-strip case analysis; the foot-exponent bound makes every retained valuation finite over any field, including the charged finite-order regime. No choice principle is used: all sums, determinants, strips and currents are finite.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Cyclotomic Schur Jantzen sum formula from rim-hook valuations

Statement

Let $F$ be any field, let $q_{H} \in F^{\times}$ have finite multiplicative order $e \geq 2$ , let $ℓ \geq 1$ be arbitrary and let $Q_{a} = q_{H}^{c_{a}}$ with charges $c_{a} \in Z$ . Fix $n \geq 0$ and let $S_{F}$ be the cyclotomic $q$ -Schur algebra over $F$ with summands indexed by all $ℓ$ -multipartitions of $n$ . Put $O = F [t, t^{- 1}]_{(t - 1)}, π = t - 1, K = F (t), z = q_{H} t, U_{a} = Q_{a} t^{n a} .$ Let $Δ_{O} (λ)$ be the standard Weyl lattice with its actual cellular form, and define $Δ_{O} (λ) (s) = {v : ⟨ v, Δ_{O} (λ) ⟩ \subseteq π^{s} O}, Δ_{F} (λ)^{s} = (Δ_{O} (λ) (s) + π Δ_{O} (λ)) / π Δ_{O} (λ) .$ Then $(Δ_{F} (λ)^{s})_{s \geq 0}$ is a filtration by $S_{F}$ -submodules, its first term is $rad Δ_{F} (λ)$ and the filtration terminates, and in the integer Grothendieck group of finite-dimensional $S_{F}$ -modules, $\sum_{s > 0} [Δ_{F} (λ)^{s}] = \sum_{μ} J_{λ μ} [Δ_{F} (μ)] .$ For a node $x = (i, j, a)$ define the rim hook $r_{x}^{λ}$ and its foot $f_{x}^{λ} = (λ_{j}^{'}, j, a)$ as in Lyle–Mathas, $leg (r_{x}^{λ}) = λ_{j}^{'} - i$ , and ${res}_{O} (i, j, a) = U_{a} z^{j - i}$ . If $λ$ strictly dominates $μ$ , set $J_{λ μ} = \sum_{\begin{array}{c} x \in [λ], y \in [μ] \\ [λ] ∖ r_{x}^{λ} = [μ] ∖ r_{y}^{μ} \end{array}} (- 1)^{leg (r_{x}^{λ}) + leg (r_{y}^{μ})} ν_{π} ({res}_{O} (f_{x}^{λ}) - {res}_{O} (f_{y}^{μ})),$ and set $J_{λ μ} = 0$ otherwise. Equal remainder diagrams force equal hook lengths. Every displayed difference is nonzero in $K$ , including same-component terms with distinct labels, by the foot-exponent separation argument; no level-one or characteristic-zero restriction is present. The same statement holds for the right Schur standard modules, by applying the argument to the opposite algebra $S^{o p}$ (The opposite ring $R^{o p}$ ): $a^{o p}$ acts by right multiplication by $a$ and $e_{ν} M$ becomes $M e_{ν}$ , with the contravariant form unchanged. At rank zero the sole form is $1$ and the asserted sum is $0$ directly, with no word-separation hypothesis needed.

Facts & Assumptions

Given: The actual arbitrary-level $q$ -Schur construction [F1], the monic Gram product [F2], the strip recursion and marked-strip commutator [F3], the DVR Jantzen leaves [F4], and the Laurent valuation leaf [F5].

[F1]

The actual multipartition construction: $S_{O}$ is finite free and cellular, $Δ_{O} (λ)$ is finite free with its contravariant cellular form, every specialized standard has a nonzero unit Gram entry $1$ , the multipartition weight projections $e_{ν}$ are selfadjoint and act on the actual semistandard type- $ν$ basis; over $K$ the Hecke and Schur standards are split simple, so their forms are nondegenerate; the standard classes form a $Z$ -basis of $K_{0}$ and their multipartition-weight Kostka submatrix is unitriangular (Cyclotomic Schur standards and corner functor control block linkage, Generic cyclotomic Schur deformation standards have nondegenerate forms, Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules).

[F2]

The proved local monic Gram product: with the actual Specht Murphy form, the dominance-minimal refinement $t_{T}$ , the row symmetrizer coefficient $C_{t_{T}}$ and the prefix polynomial eigenvalue $u_{T}$ , $\det G_{λ ν} = \prod_{T \in {SStd}_{ν} (λ)} \frac{γ_{t_{T}}}{C_{t_{T}} u_{T}} .$ The inverse-permutation-operator proof inserts the essential $m_{ν}$ inverse; the spectral transition has determinant exactly $1$ and all denominators are nonzero in the permitted $F (t)$ (Cyclotomic Schur weight Gram determinants are products of monic semistandard norms).

[F3]

The proved local pair-factor/strip recursion and the complete marked-strip identity: adding the next type-row label to a shape $ρ$ gives a permitted strip $ρ \to λ$ with factor $F_{λ ρ}^{(a)}$ , whose product over a tableau chain is the norm of [F2]; and the signed hook matrix $J$ , with all auxiliary ranks using the same fixed $z, U_{a}$ , and the coefficient-one strip matrix $Γ_{a}$ obey $(J_{n} Γ_{a}^{(n, m)} - Γ_{a}^{(n, m)} J_{m})_{λ ρ} = Γ_{a}^{(n, m)} (λ, ρ) ν_{π} (F_{λ ρ}^{(a)}),$ on support and off support (the same-component cases and the crosscomponent rank-one adjugate cases) (Cyclotomic Schur weight Gram valuations equal signed rim-hook Kostka sums).

[F4]

The DVR leaves: the Jantzen filtration is contravariant-stable, its first term is the reduced-form radical, equal to the module radical for these nonzero-form cellular standards, Smith reduction gives finite termination and $\sum_{s > 0} \dim e_{ν} Δ_{F} (λ)^{s} = V (\det G_{λ ν})$ , and exactness of the selfadjoint weight projection together with the unitriangular multipartition Kostka matrix gives integer $K_{0}$ character injectivity (DVR cellular Jantzen filtration and exact Gram valuation, Selfadjoint weight Gram valuations detect Jantzen classes, Grothendieck group of an essentially small abelian category).

[F5]

For a nonzero Laurent difference $A t^{u} - B t^{v}$ , unequal constants give valuation $0$ , while equal constants with $u - v \neq 0$ give valuation $1$ in characteristic zero and $p^{v_{p} (∣ u - v ∣)}$ in characteristic $p$ (Laurent binomial valuation at one over any field).

Proof

technique · direct

Given: The arbitrary field, level, charges, actual Schur construction and deformation specified in the Statement.

1.1givenF1F4

Filtration and radical. For $n = 0$ there is one empty multipartition, $S_{O} = O$ , $Δ_{O} = O$ with form $1$ , every positive filtration term is $0$ , and every $J$ coefficient is $0$ , so the assertions follow directly. For $n \geq 1$ , [F1] gives free standard lattices with nondegenerate generic forms. The Smith reduction of [F4] shows that the defined filtration is by $S_{F}$ -submodules, begins at $Δ_{F} (λ)$ , has first term equal to the form radical, and is zero for all sufficiently large $s$ . The rank-one cell actions of the nonzero cellular form force every proper submodule into that radical, so the first term is the module radical.

1.2F1F3

The auxiliary character identity. Introduce ordered variables for the type-row labels; the auxiliary generating series of flagged semistandard tableaux is the product over physical components $a$ of $s_{λ^{(a)}} (Y_{a})$ , where $Y_{a}$ is the alphabet of type components at least $a$ , and its coefficient at an actual multipartition $ν$ is $K_{λ ν}$ . Successive $Γ_{a}$ matrices add the horizontal strips carrying each next label and therefore enumerate these tableaux once; the path/Jacobi–Trudi cancellation proves this character identity locally. No actual Schur weight or module for another composition is needed.

2.1F2F4step 1.1

The weight-character reduction. Fix an actual multipartition weight $ν$ . The selfadjoint orthogonal decomposition by $e_{ν}$ makes the standard weight space orthogonal to every other weight space, so [F4] gives $\sum_{s > 0} \dim_{F} e_{ν} Δ_{F} (λ)^{s} = V (\det G_{λ ν})$ . By [F2] and additivity of the valuation this is the sum over type- $ν$ semistandard $T$ of the valuations of their explicit monic norms; empty weight spaces use determinant $1$ and contribute $0$ .

2.2F3step 1.2

The beta form of the hook side. Pad the beta lists to $N > n$ . Removing a hook of length $h$ lowers one beta bead by $h$ with sorting sign $(- 1)^{leg}$ , and adding it raises a bead by $h$ with the corresponding sign; the foot content is the lowered bead position minus $N$ plus $1$ , up to the unit foot-residue shift. Thus the full signed equal-length hook side is precisely the finite shifted-beta Jacobi–Trudi polynomial: same-component inward ranges, crosscomponent earlier-to-later ranges, occupied-position zero terms and the unit shift are all covered by [F3]. Its coefficient at $ν$ is $\sum_{μ} J_{λ μ} K_{μ ν}$ .

3.1F2F3step 2.2

Telescoping the marked-strip commutator. Apply the complete local identity [F3] to the product of successive type-row $Γ$ matrices and telescope $J Γ - Γ J$ , starting from the empty shape where $J_{0} = 0$ . Each commutator inserts exactly the valuation of the strip factor at that label position, while all other $Γ$ factors retain the same tableau chain; summing the inserted positions gives the total valuation of the product of the factors over the chain, hence of the monic norm by [F2] and [F3]. The left endpoint of the telescope is exactly the auxiliary polynomial of step 2.2, so at the actual multipartition coefficient $ν$ , $ν_{π} (\det G_{λ ν}) = \sum_{μ} J_{λ μ} K_{μ ν}$ . Every auxiliary rank keeps the final-rank- $n$ parameters $U_{a} = Q_{a} t^{n a}$ , with no rank-dependent deformation substituted during the telescope.

4.1F3F5step 3.1

Exactness of the valuations. For a nonzero difference $A t^{u} - B t^{v}$ , unequal constants give valuation $0$ , and equal constants with $u - v \neq 0$ give valuation $1$ in characteristic zero or $p^{v_{p} (∣ u - v ∣)}$ in characteristic $p$ by [F5]. Artificial padded symbols in the crosscomponent minor proofs cancel formally before evaluation, and the retained terms are actual valid hook or node pairs whose exponent bound makes their differences nonzero in $K$ ; hence step 3.1 is valid over every $F$ and retains the higher positive-characteristic multiplicities.

5.1F1F4step 2.1step 3.1∎

The integer Grothendieck class identity. Combining steps 2.1 and 3.1 shows that the actual multipartition-weight character of $\sum_{s > 0} [Δ_{F} (λ)^{s}]$ equals that of $\sum_{μ} J_{λ μ} [Δ_{F} (μ)]$ . Weight projection is exact, so these are additive characters on the integer $K_{0}$ group [F4], and the standard classes form its $Z$ -basis by [F1]; their multipartition-weight matrix is unitriangular, hence invertible over $Z$ . A zero difference of weight characters therefore has all standard coefficients zero, which proves the asserted integer $K_{0}$ equality, with no rationalization and no Schur-functor injectivity claim.

Remarks

The theorem is the Jantzen sum formula for the cyclotomic $q$ -Schur algebra in the exact form needed by the block theory: the filtration is defined by the integral cellular form, its radical identification and termination are DVR leaves, and the character of the positive part is the signed rim-hook valuation matrix acting on standard classes. No choice principle is used: all lattices, filtrations, and coefficient sums are finite or finitely generated over the DVR.

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Cyclotomic Jantzen equivalence is Schur and Hecke block linkage

Statement

Over any field $F$ , at any level $ℓ \geq 1$ , with $q_{H} \neq 1$ of finite order $e \geq 2$ and charges $Q_{a} = q_{H}^{c_{a}}$ , use the actual multipartition-summand Schur cover and standards of Cyclotomic Schur standards and corner functor control block linkage and the integer $J_{λ μ}$ of Cyclotomic Schur Jantzen sum formula from rim-hook valuations. Let $\sim_{J}$ be the equivalence relation on multipartitions generated by $J_{λ μ} \neq 0$ or $J_{μ λ} \neq 0$ . Then $λ \sim_{J} μ ⟺ Δ^{λ}, Δ^{μ} lie in one Schur block ⟺ S^{λ}, S^{μ} lie in one Hecke block .$ For rank $0$ there is the single empty class. No residue-content block assertion is used as an input.

Facts & Assumptions

Given: The Schur/Specht block correspondence and standard $K_{0}$ basis [F1], the terminating radical-first Jantzen formula [F2], the block convention [F3] and the Grothendieck basis convention [F4].

[F1]

Standard block purity, shared-simple-factor linkage, the unitriangular standard $K_{0}$ basis, and the Schur/Specht block correspondence: two standards $Δ^{λ}, Δ^{μ}$ share a block exactly when the corresponding Murphy Specht modules $S^{λ}, S^{μ}$ share a block, and cell modules share a block exactly when linked by a finite chain of cell modules whose neighbouring terms share a simple factor (Cyclotomic Schur standards and corner functor control block linkage).

[F2]

The Jantzen filtration $(Δ_{F} (λ)^{s})$ is radical-first and terminating, with $\sum_{s > 0} [Δ_{F} (λ)^{s}] = \sum_{μ} J_{λ μ} [Δ_{F} (μ)]$ in the integer Grothendieck group (Cyclotomic Schur Jantzen sum formula from rim-hook valuations).

[F3]

Blocks are the primitive-central-idempotent blocks (Blocks of a finite-dimensional algebra via primitive central idempotents).

[F4]

The Grothendieck group has the standard classes as a $Z$ -basis because the decomposition matrix to simple classes is triangular with diagonal $1$ (Grothendieck group of an essentially small abelian category).

Proof

technique · direct

Given: The data and suppliers of [F1]–[F4].

1.1F1F4

Standard classes form a $Z$ -basis. The matrix of standard classes to simple classes is triangular with diagonal one by [F1], hence [F4] makes the classes $[Δ (λ)]$ a $Z$ -basis of the integer Grothendieck group of the Schur cover, and the classes of the block projections are independent as well.

2.1F1F2F3step 1.1

Nonzero $J$ -coefficients link the same block. Suppose $J_{λ μ} \neq 0$ and project the Jantzen sum identity of [F2] to all blocks other than the block of $Δ (λ)$ . Its left side is zero because each $Δ_{F} (λ)^{i}$ is a submodule of the block-pure standard $Δ (λ)$ ; its right side is a $Z$ -linear combination of standard classes lying in those other blocks, and the coefficient of $[Δ (μ)]$ in it is exactly $J_{λ μ} \neq 0$ . Basis independence from step 1.1 forces that coefficient to vanish, a contradiction unless $Δ (μ)$ lies in the block of $Δ (λ)$ .

2.2F1F2F3step 1.1

Nonzero decomposition numbers force a $J$ -chain. Put $d_{λ μ} = [Δ (λ) : L (μ)]$ . For $λ \neq μ$ , the radical $Δ (λ)^{1} = rad Δ (λ)$ contains every composition factor $L (μ)$ of $Δ (λ)$ , and all higher filtration terms lie in that radical, so the coefficient of $[L (μ)]$ in the left side of the Jantzen identity is nonzero exactly when $d_{λ μ} \neq 0$ ; on the right it is $\sum_{ν} J_{λ ν} d_{ν μ}$ . If $d_{λ μ} \neq 0$ , this integer is nonzero, so some $ν$ satisfies $J_{λ ν} \neq 0$ and $d_{ν μ} \neq 0$ , with $λ ⊳ ν ⊵ μ$ . Repeating strictly decreases the index in the finite dominance poset until $ν = μ$ , so $λ \sim_{J} μ$ . Consequently any two standards sharing a simple factor are both $J$ -connected to that factor's index, and therefore to each other.

3.1F1step 2.1step 2.2∎

Blocks and Hecke linkage. By [F1], two standards share a Schur block exactly when they are linked by a finite chain of standards whose neighbouring terms share a simple factor. Step 2.2 converts each such shared factor into a $J$ -edge to the common index, and step 2.1 shows every $J$ -edge lies inside one block; hence sharing a Schur block is equivalent to $\sim_{J}$ -connectivity. Applying the established Schur/Specht block correspondence of [F1] transfers this equivalence to the Murphy Specht modules and the Hecke algebra. In rank $0$ there is one empty standard, all positive filtration terms vanish, and the single empty class is its own Jantzen class; the argument never reduces $J$ coefficients modulo the field characteristic.

Remarks

The lemma proves the Lyle–Mathas block-linkage description of the Jantzen equivalence from the local Jantzen sum formula rather than citing it: nonzero decomposition numbers propagate along strictly decreasing dominance chains, and the cellular block purity of the standards supplies the converse. No choice principle is used: all chains, coefficient sums and block projections are finite.

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Multipartitions with fixed component cores are cyclotomic Jantzen-connected

Statement

Under the arbitrary-field finite- $e$ nondegenerate cyclotomic hypotheses of Cyclotomic Jantzen equivalence is Schur and Hecke block linkage, any two $ℓ$ -multipartitions of a common rank $n$ with equal componentwise $e$ -cores are $J$ -equivalent, even if their component sizes differ. At fixed component sizes, the full-level coefficient for a change in a single component equals its level-one coefficient. The proof consumes the actual same- $e$ -core Hecke Specht block theorem for all ranks (Type-A Hecke Specht modules are in the same block iff their labels have the same e-core) rather than a readiness receipt, and imports no Carter-partition classification.

Facts & Assumptions

Given: The Jantzen linkage equivalence [F1], the exact $J$ formula [F2], the level-one identification [F3], the same-core Hecke block theorem [F4], and the $e$ -core/e-weight conventions [F5]–[F7].

[F1]

$λ \sim_{J} μ$ iff the corresponding Schur standards share a block iff the Murphy Specht modules share a Hecke block, at any level and over any field in the stated regime (Cyclotomic Jantzen equivalence is Schur and Hecke block linkage).

[F2]

The integer $J_{λ μ}$ is the signed sum of $ν_{π} ({res}_{O} (f_{x}^{λ}) - {res}_{O} (f_{y}^{μ}))$ over equal-remainder equal-length hook pairs, $0$ unless $λ$ strictly dominates $μ$ , and every foot difference is nonzero in $K$ (Cyclotomic Schur Jantzen sum formula from rim-hook valuations).

[F3]

At level one the arbitrary-level cyclotomic Hecke algebra is the finite type-A Hecke algebra (Cyclotomic Hecke algebras of level one are finite Hecke algebras).

[F4]

Two type-A Hecke Specht modules are in the same block exactly when their labels have the same $e$ -core (Type-A Hecke Specht modules are in the same block iff their labels have the same e-core).

[F5]

$e$ -core, $e$ -quotient and $e$ -weight of a partition (The e-core, the e-quotient, and the e-weight), with the core independent of the removal order (The e-core is independent of the rim hook removal order) and the size/weight formula (The core-quotient size formula defines the weight).

[F6]

The actual universal signed-conjugate comparison gives $P (λ^{T}) ≅ C_{M} (λ)^{θ}$ , hence $C_{M} (λ) ≅ P (λ^{T})^{θ}$ ; $θ$ is an involutive algebra automorphism whose pullback is exact (Ordinary Specht modules are the signed conjugate Murphy cells, The Hecke sign automorphism, pullback and contragredient dual).

Proof

technique · direct

Given: The rank $n$ , the charges, and the suppliers of [F1]–[F5].

1.1F1F3F4F5F6

One component at a time, at level one. Suppose first that the component sizes agree. For a single component $a$ of rank $m$ , the actual level-one Hecke algebra is the finite Hecke algebra by [F3]. Its same-core Specht-block clause [F4], together with the level-one case of the Jantzen linkage equivalence [F1], connects any two rank- $m$ partitions with equal $e$ -core by level-one Jantzen edges. The exact signed-conjugate bridge is $C_{M} (λ) ≅ P (λ^{T})^{θ}$ , where $θ (T_{i}) = - z T_{i}^{- 1}$ and $P$ denotes the ordinary Dipper–James Specht module. If $λ, μ$ have equal $e$ -cores, their transposes have the equal transposed cores, so [F4] puts $P (λ^{T})$ and $P (μ^{T})$ in the same ordinary Hecke block. An algebra automorphism sends primitive central idempotents bijectively to primitive central idempotents; its exact module pullback therefore preserves the equivalence of sharing a block. Applying the SAME $θ$ to both modules gives the required Murphy block equality and then [F1] gives the level-one Jantzen connection. No untwisted conjugate-dual identity or algebraic-closure assumption is used. The rank $m = 0$ case has a single empty partition and needs no block theorem.

1.2F2F5algebra

Equalizing the component weights with charged runners. Choose a component $a$ whose weight exceeds its target and a component $b$ whose weight is deficient. A positive-weight component has a removable rim $e$ -hook by [F5]. For each component $c$ use the charged infinite beta set $B_{c} = {λ_{j}^{(c)} + c_{c} - j : j \geq 1}$ . This is the ordinary beta display shifted to charge $c_{c}$ , with all sufficiently negative positions occupied and all sufficiently positive positions vacant. A removable hook is a bead move $u \mapsto u - e$ to a gap, and its charged foot residue is $u - e + 1$ modulo $e$ . To see the foot formula, order the beta positions before and after the move: the affected consecutive rows form the boundary strip between the starting and ending positions, and the leftmost removed box in the last affected row has charged content one more than the ending position. The inverse move wraps a hook whose foot has charged residue one more than its starting position. Let $f$ be the charged foot residue of the hook removed from $a$ . In $B_{b}$ , choose the largest occupied integer $v \equiv f - 1 (m o d e)$ ; it exists by the two tail conditions, and $v + e$ is vacant by maximality. Move $v$ to $v + e$ in component $b$ . Its new hook has charged foot residue $v + 1 = f$ . Equivalently, in an uncharged display for component $b$ the runner to use is $f - c_{b} - 1$ modulo $e$ , so different component charges cannot be ignored. Both moves stay on their own runners and preserve the component cores and total rank. The earlier of $a, b$ determines the strict dominance orientation: removing boxes there and adding them later decreases every intervening prefix sum; adding boxes there reverses the inequality. In that orientation the equal-remainder hook pair is unique, since the two changed components determine both strips. Its charged foot residues agree, so the foot difference evaluates to zero at $t = 1$ ; it is nonzero over $F (t)$ by [F2], and consequently has a finite positive valuation. Thus the single signed contribution is nonzero, giving a Jantzen edge. If $δ_{c}$ is the current component weight minus its target, then $δ_{a} \geq 1$ , $δ_{b} \leq - 1$ , and this transfer changes $\sum_{c} δ_{c}^{2}$ by $- 2 δ_{a} + 2 δ_{b} + 2 \leq - 2$ . Finitely many such transfers therefore equalize all component weights.

2.1F1F2step 1.1

Fixed component sizes. For two full multipartitions that differ only in component $a$ , every contributing equal-remainder hook pair in $J$ lies in $a$ : the other components are unchanged on both sides and cancelling hooks between two unchanged components would return the same multipartition. The common residue factor $Q_{a} t^{n a}$ is a unit of $O$ , and replacing it by the level-one factor $Q_{a} t^{m}$ multiplies each foot difference by a Laurent unit, so the valuation is unchanged and the full-level $J$ coefficient equals the level-one coefficient of [F2]. Inserting the level-one paths of step 1.1 in component $a$ and leaving the other components fixed connects all multipartitions with equal component sizes and the same component cores.

3.1F1F2step 2.1step 1.2∎

Conclusion. Steps 1.2 equalize component sizes along Jantzen edges from any starting multipartition, and step 2.1 connects all multipartitions of fixed component sizes with equal component cores. Composing the two finite chains connects any two multipartitions of rank $n$ with equal componentwise $e$ -cores, proving that they are $J$ -equivalent; the fixed-size comparison of coefficients is step 2.1.

Remarks

The lemma reduces the multipartition block question to the componentwise $e$ -core data and the level-one Hecke theorem, using the exact level-one to finite-Hecke identification and the explicit hook transfer between components. No choice principle is used: all chains and hook moves are finite and explicitly constructed.

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Cyclotomic multipartitions have equal residue content exactly when Jantzen-connected

Statement

Let $F$ be any field, let $q_{H} \in F^{\times}$ have finite order $e \geq 2$ with $q_{H} \neq 1$ , let $ℓ \geq 1$ and $Q_{a} = q_{H}^{c_{a}}$ . For all $ℓ$ -multipartitions $λ, μ$ of rank $n \geq 0$ , the node-residue contents $C_{i} (λ) = # {(r, s, a) \in [λ] : c_{a} + s - r \equiv i (m o d e)}$ agree for every $i \in Z / e Z$ if and only if $λ \sim_{J} μ$ for the exact integer Jantzen coefficients of Cyclotomic Schur Jantzen sum formula from rim-hook valuations. This includes arbitrary levels and all field characteristics.

Facts & Assumptions

Given: The exact $J$ formula [F1], the charged core-matrix connectivity [F2], the fixed-component-core lift lemma [F3], and the multipartition residue conventions [F4].

[F1]

$J_{λ μ}$ is the signed integer sum of $ν_{π} ({res}_{O} (f_{x}^{λ}) - {res}_{O} (f_{y}^{μ}))$ over equal-remainder equal-length hook pairs, $0$ unless $λ$ strictly dominates $μ$ (Cyclotomic Schur Jantzen sum formula from rim-hook valuations).

[F2]

Charged core height matrices admit content-preserving descent and minimal switch connectivity: within a content class of rank- $\leq n$ cores with the same hub there is a sequence of $γ = 2$ zero-size switches followed, unless already minimal, by a $γ \geq 3$ strict descent, every intermediate core having size at most $n$ and the same hub (Charged core height matrices: hub-preserving descent and minimal switch connectivity).

[F3]

Multipartitions of rank $n$ with equal componentwise $e$ -cores are $J$ -equivalent, and the full-level coefficient for a change in one fixed-size component equals its level-one coefficient (Multipartitions with fixed component cores are cyclotomic Jantzen-connected).

[F4]

Multipartitions, their nodes $(r, s, a)$ , the charged parameters $z, U_{a}$ and residue conventions (Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules).

Proof

technique · direct

Given: The rank $n$ , the charges $c_{a}$ , and the suppliers of [F1]–[F4].

1.1F1F4algebra

Necessity. Suppose $J_{λ μ} \neq 0$ ; then at least one term of its signed integer sum has nonzero valuation, and evaluating that generic foot difference at $t = 1$ gives zero, so the two foot eigenvalues coincide and, since $q_{H}$ has order $e$ , their charged foot residues agree. The two hooks have equal length and the same remainder diagram. Along a rim strip, reading from its lowest-left node to its highest-right node increases the content by one at every step, so its residue multiset consists of $h$ consecutive residues starting at its foot; hence the two removed hooks have identical residue counts and $C (λ) = C (μ)$ . Transitivity gives the necessity of $C (λ) = C (μ)$ for every $J$ -path.

1.2F2F4

Sufficiency setup: cores and lifts. Let $κ$ be the componentwise core of a member of the rank- $n$ content class; its size is at most $n$ and congruent to $n$ modulo $e$ , and by [F2] all such cores have the same hub and are connected by the prescribed switch sequence. Every intermediate core has size at most $n$ and the same hub, so adding $(n - core size) / e$ $e$ -hooks produces a rank- $n$ multipartition of the same content by the equal-hub/equal-size clause of [F2].

2.1F1F2step 1.2

The explicit one-hook lift across a core switch. Let $κ^{'}$ be obtained from $κ$ by the oriented switch in rows $i, j$ and components $a, b$ of [F2], with $γ \geq 2$ , so $∣ κ^{'} ∣ = ∣ κ ∣ - e D$ with $D = γ - 2 \geq 0$ . Write $u = i + e t_{i a}$ , $v = j + e (t_{j a} + 1)$ , $v^{'} = j + e t_{j b}$ , $u^{'} = i + e (t_{i b} + 1)$ : the switch removes bead $u$ and inserts $v$ in component $a$ , and removes $v^{'}$ and inserts $u^{'}$ in component $b$ . Put $w = (n - ∣ κ ∣) / e \geq 0$ , form $λ$ from $κ$ by moving the lowest bead $u$ of the $i$ -runner in component $a$ to $u + e w$ (adding $w$ $e$ -hooks on this runner), and form $μ$ from $κ^{'}$ by moving its lowest bead $v$ on the $j$ -runner in component $a$ to $v + e (w + D)$ . Both have size $n$ , the same hub, and by [F2] equal content. Their beta sets in component $a$ differ only by replacing $u + e w$ with $v + e (w + D)$ , and those in component $b$ only by replacing $v^{'}$ with $u^{'}$ . Because $(v - u) + (u^{'} - v^{'}) = - e D$ , we have $v + e (w + D) - (u + e w) = v^{'} - u^{'}$ , so the two components exchange a single rim hook of length $∣ v^{'} - u^{'} ∣$ , nonzero since $i \neq j$ modulo $e$ . If $v^{'} - u^{'} > 0$ , component $b$ unwraps and $a$ wraps a hook with foot residue $i + 1$ modulo $e$ ; otherwise $a$ unwraps and $b$ wraps one with foot residue $j + 1$ . In both cases the hook feet have equal residues, and the hook pair is uniquely determined because distinct components change, so the $J$ coefficient of [F1] is a single signed valuation, a nonzero integer with no cancellation. Hence $λ$ and $μ$ are Jantzen-connected by one edge; this supplies the detail abbreviated in the Lyle–Mathas tight lemma.

3.1F1F2F3step 1.1step 2.1∎

Termination and conclusion. The generic difference of the unique hook contribution is nonzero by [F1], and its specialization vanishes because the charged feet agree, so its positive integer valuation cannot disappear under reduction modulo $char F$ . The fixed-core lemma [F3] permits replacing arbitrary lifts by the chosen lifts before and after each switch, and strict descent lowers the nonnegative core size by at least $e$ , so the process terminates at a minimum core; all minimum cores have equal size and are connected by $γ = 2$ switches, whose rank- $n$ lifts are connected by the same explicit construction of step 2.1. Together with the fixed-component-size lifts of [F3], every member of the content class is Jantzen-connected to every other, and step 1.1 gives the converse. The empty rank has only one multipartition and is immediate; no affine-center or primitive-block theorem has entered the proof.

Remarks

The theorem is the exact content classification of the Jantzen classes: equal charged node-residue content is equivalent to connectivity by the integer rim-hook coefficient matrix, uniformly in the characteristic and the level. The construction of the lifts and the termination argument are finite and explicit. No choice principle is used.

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Central content sectors are zero or primitive cyclotomic Hecke blocks

Statement

Let $F$ be any field and let $q_{H} \in F^{\times}$ have finite quantum characteristic $e \geq 2$ . Use the standard cyclotomic type-A Hecke presentation with level $ℓ = \sum_{i} λ_{i} \geq 0$ imposed on the first polynomial variable, with $λ_{i} \geq 0$ and $I = Z / e Z$ . For $q_{H} \neq 1$ the roots are $q_{H}^{i}$ and $q_{H}$ has order $e$ ; for $q_{H} = 1$ one has $char F = e > 0$ and the roots are the prime-field residues $i$ . Using the central CRT notation of Central content sectors of the cyclotomic Hecke algebra and Polynomial spectra and central content sectors of finite cyclotomic Hecke algebras, for every content $α$ of rank $d$ the element $e_{α} = \sum_{word content α} e (word)$ is either zero or a primitive central idempotent. The nonzero $e_{α}$ are exactly all primitive central idempotents, so $H_{d}^{Λ} = ⨁_{α} e_{α} H_{d}^{Λ}$ is precisely the block decomposition. At $d = 0$ the unique sector is $F$ , regardless of level; at $d > 0$ , $ℓ = 0$ the algebra is zero. This is an arbitrary-level result over the original field in both branches.

Facts & Assumptions

Given: The central content-sector notation and CRT [F1], the block convention [F2], the degenerate center identification [F3], the primitive split content idempotents of symmetric polynomial centers [F4], the Murphy cellular basis and finite cellular heads [F5]–[F6], the Jucys–Murphy content triangularity [F7], the fixed-content connectivity theorem [F8], and the Jantzen/block linkage lemma [F9].

[F1]

The pairwise-orthogonal joint residue idempotents $e (w o r d)$ sum to $1$ , the sums $e_{α}$ are central and orthogonal, and the regular/module CRT interpretation holds; no primitivity is asserted (Central content sectors of the cyclotomic Hecke algebra, Polynomial spectra and central content sectors of finite cyclotomic Hecke algebras).

[F2]

Blocks are the primitive-central-idempotent blocks (Blocks of a finite-dimensional algebra via primitive central idempotents).

[F3]

In the degenerate branch $q_{H} = 1$ the center is the image of the symmetric polynomials (The degenerate cyclotomic Hecke center is the symmetric polynomial image).

[F4]

Symmetric polynomial centers have primitive split content idempotents, applying to the early CRT projectors without a field extension or a power-sum generation assertion (Symmetric polynomial centers have primitive split content idempotents).

[F5]

The arbitrary-level Murphy elements are a cellular basis with the higher two-sided ideals (The arbitrary-level cyclotomic Hecke Murphy basis is cellular), so the nonzero cellular heads $D^{λ}$ are all the simple modules (Nonzero-form cellular heads, cell purity and shared-factor linkage).

[F6]

Every word on $S^{λ}$ , and hence on its quotient $D^{λ}$ , is the node content $C (λ)$ (Arbitrary-level Murphy Jucys–Murphy action is content triangular).

[F7]

Multipartitions have equal node-residue content exactly when they are $J$ -connected (Cyclotomic multipartitions have equal residue content exactly when Jantzen-connected).

[F8]

$J$ -connectivity is equivalent to sharing a Hecke block (Cyclotomic Jantzen equivalence is Schur and Hecke block linkage).

Proof

technique · direct

Given: The field $F$ , the finite quantum characteristic $e$ , the

multiplicities $λ_{i}$ , and the suppliers of [F1]–[F8].

1.1F1

Base cases. At rank $0$ the algebra is $F$ , the empty-word projector is $1$ , and this is its unique primitive central idempotent. At positive rank and level $0$ the defining polynomial is $1$ , so the algebra is $0$ and every content sector is $0$ , with no nonzero blocks. Hence assume rank $d \geq 1$ and level $ℓ \geq 1$ .

1.2F1F3F4

The degenerate branch. In the branch $q_{H} = 1$ necessarily $char F = e > 0$ . The center is the full symmetric-polynomial image over this $F$ by [F3], and [F4] applies to the early CRT projectors and proves that their content sums are zero or primitive. This uses neither a field extension nor a power-sum generation assertion.

1.3F1F5F6

The nondegenerate branch: labels of sectors. In the branch $q_{H} \neq 1$ choose a charge list $c_{1}, \dots, c_{ℓ}$ repeating each $i$ exactly $λ_{i}$ times, so $Q_{a} = q_{H}^{c_{a}}$ and the first polynomial is unchanged. The arbitrary-level Murphy cellular basis and the finite cellular head lemma [F5] supply all simple $H_{d}^{Λ}$ -modules as nonzero Specht quotients $D^{λ}$ , and the content-triangular JM action [F6] says every word on $S^{λ}$ , hence on $D^{λ}$ , has the node content $C (λ)$ . Thus a simple module lying in a nonzero sector $e_{α}$ has a label $λ$ with $C (λ) = α$ .

2.1F1F7F8step 1.3

Primitivity of the nonzero sectors. Any two simple modules in the sector $e_{α}$ have labels with equal content, so [F7] connects the labels by integer Jantzen edges and [F8] transfers this to a common Hecke block. If $e_{α}$ decomposed into two nonzero orthogonal central idempotents $u, v$ , the nonzero finite algebras $u H_{d}^{Λ}$ and $v H_{d}^{Λ}$ would each have a simple module, obtained from a proper right ideal of maximal dimension. Those simple modules cannot lie in one block, because the central idempotent $u$ acts as the identity on the first and as zero on the second, and all modules and extensions split by $u$ . This contradicts the one-block assertion, so $e_{α}$ is primitive.

3.1F1F2step 1.2step 2.1∎

Every block is a content sector. In either branch let $b$ be a primitive central idempotent and write $b = \sum_{α} b e_{α}$ . The terms are central orthogonal idempotents, so precisely one is nonzero and $b = b e_{α}$ ; primitivity of the nonzero $e_{α}$ from step 1.2 or step 2.1 gives $b = e_{α}$ . Hence the nonzero content sums are exactly all primitive central idempotents and the decomposition $H_{d}^{Λ} = ⨁_{α} e_{α} H_{d}^{Λ}$ is precisely the block decomposition [F2]. All sums are finite, the field characteristic never reduces an integer $K_{0}$ adjacency, and no affine-block classification or citation-only center theorem is consumed.

Remarks

The theorem identifies the primitive central idempotents of the arbitrary-level cyclotomic Hecke algebra with the nonzero content sectors, in both the degenerate and nondegenerate branches and over the original field. The two branches are routed through the symmetric-polynomial center and the content-classification theorem respectively; no primitive-block theorem is assumed. No choice principle is used: all sums, sectors and simple selections are finite.

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Residue-sequence idempotents and cyclotomic Hecke blocks

Definition

Work in the charged finite- $e$ field conventions of Affine and cyclotomic type-A Hecke algebras. By Polynomial spectra and central content sectors of finite cyclotomic Hecke algebras, $H_{d}^{Λ}$ is finite-dimensional and each minimal polynomial of the polynomial generator $z_{r} = X_{r}$ (nondegenerate normalization) respectively $x_{r}$ (degenerate normalization) splits over the residues $c_{i} = q_{H}^{i}$ respectively $i \cdot 1_{F}$ .

Residue-sequence idempotents. Let $ε_{r} (i)$ be the canonical CRT primary idempotent of that generator at the spectral value $c_{i}$ associated with residue label $i$ (zero when that value is absent), and put $e (ν) = \prod_{r = 1}^{d} ε_{r} (ν_{r})$ for a residue sequence $ν = (ν_{1}, \dots, ν_{d}) \in (Z / e Z)^{d}$ . On every finite $H$ -module the element $e (ν)$ projects onto the joint generalized residue space, and these mutually orthogonal idempotents sum to $1$ . At content $α$ put $e_{α} = \sum_{ν \in I_{α}} e (ν),$ the sum over residue sequences of content $α$ ; it is central, and $H_{d}^{Λ}$ is the direct sum of its central ideals $H_{α}^{Λ} = e_{α} H_{d}^{Λ} = e_{α} H_{d}^{Λ} e_{α} .$ By Central content sectors are zero or primitive cyclotomic Hecke blocks, every nonzero $e_{α}$ is primitive, so every nonzero $H_{α}^{Λ}$ is an actual block (Blocks of a finite-dimensional algebra via primitive central idempotents); this block assertion is proved there, not retained as an unused citation. The algebra basis and dimension convention is that of Basis and dimension of the cyclotomic Hecke algebra.

Level one. At level one, $Λ = Λ_{j}$ , the polynomial generators correspond under the explicit finite-Hecke inverse of Cyclotomic Hecke algebras of level one are finite Hecke algebras to $X_{r} = q_{H}^{j} (1 + (q_{H} - 1) L_{r}) (nondegenerate), x_{r} = j + L_{r} (degenerate),$ where $L_{r}$ is the ordinary additive Jucys–Murphy element. Thus on a standard tableau node of content $m$ their eigenvalues are $q_{H}^{j + m}$ respectively $j + m$ , while $L_{r}$ itself has additive $q$ -content $[m]_{q_{H}}$ . The nonzero joint projectors are precisely the residue words of standard tableaux (The Hecke Jucys-Murphy action is dominance triangular), and grouping those words by content gives the block labels.

Boundary conventions. At rank zero the empty projector is $1$ and the unique block is $F$ ; at level zero and positive rank all projectors and blocks are zero. The projection, centrality, and block-decomposition assertions recorded here are exactly the proved content-sector statements of the linked suppliers; this definition introduces no new primitive-block classification of its own. It uses no choice principle: all spectra, projectors, and sums are finite.

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The Brundan–Kleshchev blockwise Hecke–KLR isomorphism

Statement

Let $F$ be any field, let $q \in F^{\times}$ have finite quantum characteristic $e = e (F, q) \geq 2$ , let $Λ \in P^{+}$ be dominant and let $α \in Q_{+}$ have height $d$ with $e_{α} \neq 0$ , in the conventions of Affine and cyclotomic type-A Hecke algebras and Residue-sequence idempotents and cyclotomic Hecke blocks. Then the homomorphisms $ρ ⁣ : R_{α}^{Λ} \to H_{α}^{Λ}$ of Hecke intertwiners produce the local KLR generators and $σ ⁣ : H_{α}^{Λ} \to R_{α}^{Λ}$ of The cyclotomic KLR block satisfies the Hecke relations are mutually inverse $F$ -algebra isomorphisms. Equivalently, $H_{α}^{Λ}$ is generated by the Khovanov–Lauda generators ${e (i)} \cup {y_{r}} \cup {ψ_{r}}$ subject only to the relations (1.6)–(1.15), and the isomorphism is normalized by $ρ (e (i)) = e (i)$ and $ρ (y_{r} e (i)) = (1 - q^{- i_{r}} X_{r}) e (i) (q \neq 1), ρ (y_{r} e (i)) = (x_{r} - i_{r}) e (i) (q = 1) .$ The degenerate ( $q = 1$ , $char F = e$ ) and nondegenerate ( $q \neq 1$ ) cases are proved separately, and both hold in finite quantum characteristic $e$ .

Facts & Assumptions

Given: The residue-projector/block definition [F1], the early central sector isomorphism [F2], the local KLR generator construction [F3], and the inverse Hecke-relations map [F4].

[F1]

The residue-sequence idempotents $e (ν)$ , the central sums $e_{α}$ and the direct decomposition $H_{d}^{Λ} = ⨁_{α} H_{α}^{Λ}$ , with every nonzero $e_{α}$ primitive and every nonzero $H_{α}^{Λ}$ an actual block; at level one the JM generators correspond to $X_{r} = q^{j} (1 + (q - 1) L_{r})$ respectively $x_{r} = j + L_{r}$ (Residue-sequence idempotents and cyclotomic Hecke blocks).

[F2]

The explicit maps $ρ ⁣ : R_{α}^{Λ} \to H_{α}^{Λ}$ and $σ ⁣ : H_{α}^{Λ} \to R_{α}^{Λ}$ are mutually inverse for every content $α$ , including zero sectors, with $H_{α}^{Λ} = 0$ iff $R_{α}^{Λ} = 0$ and $H_{d}^{Λ} ≅ ⨁_{α} R_{α}^{Λ}$ ; on idempotents the maps are the literal identity and the dot image is $x_{r} - i_{r}$ in the degenerate branch and $1 - q^{- i_{r}} X_{r}$ in the nondegenerate branch (Cyclotomic Hecke central sectors are isomorphic to type-A cyclotomic KLR algebras).

[F3]

The elements ${e (i)} \cup {y_{r}} \cup {ψ_{r}}$ satisfy the relations (1.6)–(1.15) and define the unique homomorphism $ρ$ with $e (i) \mapsto e (i)$ , $y_{r} \mapsto y_{r}$ , $ψ_{r} \mapsto ψ_{r}$ (Hecke intertwiners produce the local KLR generators).

[F4]

The elements $x_{r}, s_{r}$ respectively $X_{r}, T_{r}$ satisfy the affine Hecke and cyclotomic relations, so there is a unique homomorphism $σ$ killing $e_{β}$ and every corner $e_{β} H_{d}^{Λ} e_{γ}$ with $(β, γ) \neq (α, α)$ (The cyclotomic KLR block satisfies the Hecke relations).

Proof

technique · direct

Given: The field $F$ , the finite quantum characteristic $e$ , the dominant

weight $Λ$ , the content $α$ of height $d$ with $e_{α} \neq 0$ , and

the suppliers of [F1]–[F4].

1.1F2F3F4

The mutually inverse early maps. By [F3] the local KLR generators of the nonzero sector satisfy the KLR relations, so there is a unique algebra homomorphism $ρ ⁣ : R_{α}^{Λ} \to H_{α}^{Λ}$ sending $e (i) \mapsto e (i)$ , $y_{r} \mapsto y_{r}$ and $ψ_{r} \mapsto ψ_{r}$ ; by [F4] the Hecke generators $X_{r}, T_{r}$ (respectively $x_{r}, s_{r}$ ) of the sector satisfy the affine Hecke and cyclotomic relations, so there is a unique homomorphism $σ ⁣ : H_{α}^{Λ} \to R_{α}^{Λ}$ with $σ (e (i)) = e (i)$ for $i \in I_{α}$ . The early central sector isomorphism [F2] proves, by explicit generator composites that are literally the identity, that $ρ$ and $σ$ are mutually inverse over $F$ for every content, including the zero sectors, with $H_{α}^{Λ} = 0$ if and only if $R_{α}^{Λ} = 0$ . No algebraic closure, root extraction, dimension argument or primitive-block premise was used to construct these maps.

1.2F1

Primitivity of the sector. By [F1] every nonzero $e_{α}$ is primitive and every nonzero $H_{α}^{Λ}$ is an actual block. Hence for the content $α$ of the Statement the sector $H_{α}^{Λ}$ is a block of $H_{d}^{Λ}$ , and the block terminology in the Statement is justified rather than assumed.

2.1F1F2F3F4step 1.1step 1.2∎

The blockwise isomorphism and its normalization. Restricting the mutually inverse maps of step 1.1 to the nonzero sector $H_{α}^{Λ}$ gives mutually inverse unital $F$ -algebra isomorphisms $ρ ⁣ : R_{α}^{Λ} \to H_{α}^{Λ}$ and $σ ⁣ : H_{α}^{Λ} \to R_{α}^{Λ}$ . The defining formulas of $ρ$ give $ρ (e (i)) = e (i)$ and the stated dot normalization $ρ (y_{r} e (i)) = (1 - q^{- i_{r}} X_{r}) e (i)$ in the nondegenerate case and $ρ (y_{r} e (i)) = (x_{r} - i_{r}) e (i)$ in the degenerate case, matching the dot images of [F2]. Because $R_{α}^{Λ}$ is the algebra generated by ${e (i)} \cup {y_{r}} \cup {ψ_{r}}$ subject only to the defining KLR relations and $ρ$ is an isomorphism, $H_{α}^{Λ}$ is generated by the images of those generators subject only to those relations. The degenerate and nondegenerate cases were proved separately by [F2]–[F4] and both hold in finite quantum characteristic $e$ , with no characteristic-zero or level-one restriction.

Remarks

The theorem is the Brundan–Kleshchev identification of each nonzero cyclotomic Hecke block with the corresponding cyclotomic Khovanov–Lauda block. The isomorphism is exhibited by explicit mutually inverse maps, and the later primitivity theorem is used only to justify the block terminology; no choice principle is used.

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Cyclotomic Hecke blocks inherit $Z$ -gradings

Statement

In the situation of The Brundan–Kleshchev blockwise Hecke–KLR isomorphism, each fixed choice of admissible local normalising series transports the KLR homogeneous decomposition to a $Z$ -grading on $H_{α}^{Λ}$ . This is the unique grading for those fixed generator images with $\deg e (i) = 0$ , $\deg y_{r} = 2$ and $\deg ψ_{r} e (i) = - a_{i_{r}, i_{r + 1}}$ (The KLR grading and the diagrammatic multiplication). Different admissible series choices give graded-isomorphic transported algebras via $ρ^{'} \circ ρ^{- 1}$ ; their underlying identity map need not be graded. The grading can be nontrivial, but it may be trivial on an individual block, including the one-dimensional level-one rank-two $e = 3$ blocks. The ordinary Hecke presentation always has the trivial assignment of degree zero to all its generators; the transported grading is generally expressed using the nilpotent dots and local crossings, and ordinary Hecke generators can be sums of different homogeneous pieces and may be homogeneous in special cases. No claim forbids the degree-zero assignment or asserts that every block grading is nontrivial.

Facts & Assumptions

Given: The blockwise Hecke–KLR isomorphism [F1], the KLR grading conventions [F2], the cyclotomic quotient conventions [F3], and the type-A specialization conventions [F4].

[F1]

For every content $α$ with $e_{α} \neq 0$ there are mutually inverse $F$ -algebra isomorphisms $ρ ⁣ : R_{α}^{Λ} \to H_{α}^{Λ}$ and $σ$ , normalized by $ρ (e (i)) = e (i)$ and the dot images $1 - q^{- i_{r}} X_{r}$ or $x_{r} - i_{r}$ (The Brundan–Kleshchev blockwise Hecke–KLR isomorphism).

[F2]

The KLR algebra $R_{α}^{Λ}$ is graded by $\deg e (i) = 0$ , $\deg y_{r} = 2$ and $\deg ψ_{r} e (i) = - a_{i_{r}, i_{r + 1}}$ , with homogeneous local relations (The KLR grading and the diagrammatic multiplication, The cyclotomic KLR quotient at a dominant integral weight).

[F3]

The charged type-A specialization and its presentation conventions (The type-A cyclotomic specialization of the QG-KLR algebras).

Proof

technique · direct

Given: The field $F$ , the finite quantum characteristic $e \geq 2$ , the

dominant weight $Λ$ , and the suppliers of [F1]–[F3].

1.1F1F2F3algebra

Transport of the grading. Fix a choice of admissible local normalising series, hence the isomorphism $ρ$ of [F1]. Declare each image of a homogeneous KLR generator to be homogeneous of the same degree; the images generate $H_{α}^{Λ}$ and the KLR relations of [F2] are homogeneous, so the resulting degree assignment is well defined. Products retain degree addition because $ρ$ is an algebra map, so the degree summands are exactly the transported homogeneous summands. Since the generator images span the algebra, their assigned degrees determine every degree summand and give uniqueness of the grading for that fixed choice, with $\deg e (i) = 0$ , $\deg y_{r} = 2$ and $\deg ψ_{r} e (i) = - a_{i_{r}, i_{r + 1}}$ .

2.1F1F2step 1.1

Independence of the series choice. If $ρ^{'}$ is the isomorphism attached to another admissible series, then $ρ^{'} \circ ρ^{- 1}$ sends each homogeneous native generator image of the first choice to the corresponding new image and hence is graded by construction; the two transported algebras are graded-isomorphic, while the underlying identity map on $H_{α}^{Λ}$ need not be graded because the two series differ by units that need not be homogeneous of degree zero.

2.2F1step 1.1

Triviality is not excluded. Assigning degree zero to every ordinary Hecke generator makes all defining relations homogeneous, so the degree-zero assignment is a legitimate grading of the ordinary Hecke presentation and no impossibility statement is asserted. A one-dimensional sector has only degree zero, because its unit has degree zero and it equals $F$ ; in particular the one-dimensional level-one rank-two $e = 3$ blocks are trivially graded.

2.3F1F2step 1.1

A concrete nontrivial witness. Consider level one, $e = 2$ , rank two. Of the four colour words only $(0, 1)$ survives: the first-strand exponent zero kills $(1, 0)$ , and on $(0, 0)$ the equal-colour mixed identities give $ψ_{1} y_{2} = 1$ together with $ψ_{1}^{2} = 0$ , hence $ψ_{1} = 0$ after left multiplication by $ψ_{1}$ , giving $1 = 0$ ; so the equal-colour sector is zero, with $y_{1} = 0$ and the crossing zero. The quadratic double-edge relation gives $y_{2}^{2} = 0$ . Conversely $e (0, 1) = 1$ , $y_{2} = t$ , $y_{1} = ψ_{1} = 0$ in $F [t] / (t^{2})$ satisfy all rank-two relations, so that sector is $F [t] / (t^{2})$ with $\deg t = 2$ , and its Brundan–Kleshchev image supplies an actual nontrivial transported grading, including in the characteristic-two degeneration. This is a witness that the construction can be nontrivial, not a claim about all sectors.

3.1F1step 1.1step 2.1step 2.3∎

Conclusion. Each fixed admissible series choice transports the KLR homogeneous decomposition to a $Z$ -grading of $H_{α}^{Λ}$ , unique for the induced generator degrees; different choices give graded-isomorphic transported algebras; and the grading may be nontrivial on some blocks while trivial on others, with the ordinary presentation retaining its degree-zero assignment.

Remarks

The corollary records exactly what the blockwise isomorphism yields: a transported grading attached to a chosen normalising series, not a canonical intrinsic grading of the Hecke presentation, and not a claim of nontriviality on every block. No choice principle is used: the series choices and the gradings are explicit and finite.

5 · Examples, counterexamples and false statements

None yet.