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Cores, Quotients, and Blocks of Symmetric Groups

1 · Prerequisites

2 · Summary

This page develops the abacus calculus of partitions and applies it to classify the blocks of the q-Schur algebras in the stable range $d \geq n$ , the type-A Hecke algebras, and the modular symmetric groups by $e$ -cores. The computational model is fixed by Beta-sets, the shift, and the e-runner abacus of a partition and Hooks are bead-gap pairs on the abacus, where hooks become bead-gap pairs and rim-hook removal becomes a bead-to-gap relocation; hooks of $e$ -divisible length move beads along their own runners. The $e$ -core, the $e$ -quotient and the $e$ -weight are defined in The e-core, the e-quotient, and the e-weight, and The e-core is independent of the rim hook removal order proves that the terminal partition of iterated $e$ -hook removal does not depend on the order. The e-quotient controls the hooks of e-divisible length transfers the resulting dictionary to the hooks of $e$ -divisible length, and The core-quotient size formula defines the weight records the size identity $n = ∣ C_{e} (λ) ∣ + e w_{e} (λ)$ .

The second half passes from combinatorics to blocks. The linear independence of virtual Weyl combinations (Virtual Weyl combinations detect q-Schur block separation) turns the Jantzen-Schaper sum formula into the linkage criterion of q-Schur Weyl modules are in the same block iff their labels have the same e-core, with the weight-one pattern of The weight-one block decomposition pattern as the first explicit decomposition matrix. The Schur functor transports the classification to Hecke Specht modules (Type-A Hecke Specht modules are in the same block iff their labels have the same e-core), and specialisation at $q = 1$ gives Nakayama's classification of the $p$ -blocks of the symmetric groups (Nakayama block classification for symmetric groups). The weight-zero case is resolved completely in Weight-zero symmetric-group blocks are defect zero, while Block weight is not the same as numerical defect records precisely how far the combinatorial weight and the numerical defect may be compared.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Beta-sets, the shift, and the e-runner abacus of a partition

Definition

Let $λ ⊢ n$ be a partition with English Young diagram $[λ]$ , parts $λ_{1} \geq \dots \geq λ_{m} > 0$ and conjugate $λ^{'}$ (Partitions, English diagrams, and conjugation). For $1 \leq i \leq m$ put $b_{i} (λ) : = λ_{i} + m - i = h_{i, 1} (λ),$ the hook length of the first node of row $i$ (Hook, arm, leg, and hook length of a box); since $h_{i, 1} (λ) = λ_{i} + λ_{1}^{'} - i$ and $m = λ_{1}^{'}$ , the two expressions agree. Then $X_{λ} : = {b_{1} (λ) > b_{2} (λ) > \dots > b_{m} (λ)}$ is the set of first-column hook lengths of $λ$ , a finite strictly decreasing set of positive integers.

A $β$ -set is a finite set $X = {h_{1} > \dots > h_{m}} \subseteq N_{0}$ of nonnegative integers, written in strictly decreasing order. For $l \in N_{0}$ the $l$ -shift of $X$ is $X + l : = {h + l : h \in X} \cup {l - 1, l - 2, \dots, 1, 0}$ (for $l = 0$ this is $X$ itself, the second set being empty), and the partition of $X$ is $P (X) : = (h_{1} - (m - 1), h_{2} - (m - 2), \dots, h_{m} - 0),$ with trailing zeros deleted; this is a weakly decreasing sequence of nonnegative integers ending in $h_{m}$ , hence a partition of $\sum_{i} h_{i} - m (m - 1) / 2$ after deleting its zeros. A $β$ -set for $λ$ is a $β$ -set $X$ with $P (X) = λ$ .

Let $e \geq 2$ . James's $e$ -abacus has $e$ runners $0, 1, \dots, e - 1$ drawn left to right, with rows labelled $a = 0, 1, 2, \dots$ downward; the position in row $a$ of runner $i$ carries the label $a e + i$ . An $e$ -abacus configuration $A$ is a finite set of beads placed in distinct positions of this abacus; a position without a bead is a gap. The configuration $A_{X}$ of a $β$ -set $X$ carries a bead in the positions of $X$ , and conversely $X_{A}$ is the set of position labels of the beads of $A$ ; the two constructions are mutually inverse, so they biject finite $β$ -sets with finite bead configurations. Finally $P (A) : = P (X_{A})$ , and $A_{λ} : = A_{X_{λ}}$ .

Remarks

$P (X_{λ}) = λ$ . For $1 \leq i \leq m$ one has $h_{i, 1} (λ) - (λ_{1}^{'} - i) = (λ_{i} + λ_{1}^{'} - i) - (λ_{1}^{'} - i) = λ_{i}$ , so $X_{λ}$ is a $β$ -set for $λ$ ; the shift convention of the definition is exactly the one for which this holds with $l = 0$ .
Shifts describe all $β$ -sets of $λ$ . Let $X = {h_{1} > \dots > h_{m}}$ be a $β$ -set and $t = ℓ (λ)$ the number of parts of $λ$ . By the definition of $P$ and the deletion of trailing zeros, $P (X) = λ$ if and only if $m \geq t$ , $h_{j} - (m - j) = λ_{j}$ for $j \leq t$ and $h_{j} - (m - j) = 0$ for $t < j \leq m$ . This holds if and only if $m - t \in N_{0}$ and $h_{j} = (λ_{j} + t - j) + (m - t) = b_{j} (λ) + (m - t)$ for $j \leq t$ while ${h_{t + 1}, \dots, h_{m}} = {m - t - 1, \dots, 1, 0}$ ; that is, if and only if $X = X_{λ} + (m - t)$ . Hence the $β$ -sets for $λ$ are exactly the sets $X_{λ} + l$ with $l \in N_{0}$ , and $l$ is recovered as $l = m - t$ ; in particular $P (X + l) = P (X)$ for every $β$ -set $X$ and every $l$ , since the two sets have the same $P$ . This is Lemma 4.2 of the lecture notes and the display preceding (5.21) of the manuscript; it is the sense in which the shift is a normalisation and not a change of partition.
The runner labelling. Writing $h = a e + i$ with $a \in N_{0}$ and $i \in {0, 1, \dots, e - 1}$ gives a bijection $N_{0} \to N_{0} \times {0, \dots, e - 1}$ , so every position of the abacus has exactly one label and $X \mapsto A_{X}$ is a bijection onto the finite bead configurations, with inverse $A \mapsto X_{A}$ . The runner of a position $h$ is its residue $h m o d e$ ; the residue of a bead is the residue of its label. No choice principle is used anywhere in this definition.

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

Hooks are bead-gap pairs on the abacus

Statement

Let $e \geq 2$ , let $λ ⊢ n$ and let $X = {h_{1} > h_{2} > \dots > h_{m}}$ be a $β$ -set for $λ$ , so that $P (X) = λ$ (Beta-sets, the shift, and the e-runner abacus of a partition). Write $H_{i} (λ)$ for the set of hook lengths in row $i$ of $[λ]$ : thus $h \in H_{i} (λ)$ means $h = h_{i, j} (λ)$ for some node $(i, j) \in [λ]$ , and $H_{i} (λ) = \emptyset$ for $i > ℓ (λ)$ (Hook, arm, leg, and hook length of a box).

(i) For every $h \geq 0$ and every $i \in [m]$ , $h \in H_{i} (λ)$ if and only if $h \leq h_{i}$ and $h_{i} - h \notin X$ . Equivalently, the hooks of $λ$ correspond bijectively to the bead-gap pairs $(b, g)$ with $b \in X$ , $g \notin X$ and $0 \leq g < b$ ; the hook $H_{i, j} (λ)$ corresponds to $(b, g) = (h_{i}, h_{i} - h_{i, j} (λ)),$ and then $∣ H_{i, j} (λ) ∣ = b - g$ .

(ii) Fix a node $(i, j) \in [λ]$ and let $(b, g)$ be its bead-gap pair as in (i). Then $X^{'} : = (X ∖ {b}) \cup {g}$ is again a $β$ -set, now for the partition $λ ∖ R_{i, j}$ obtained from $λ$ by removing the removable border strip $R_{i, j}$ that corresponds to the hook of $(i, j)$ , of size $h_{i, j} (λ)$ and height $ℓ (i, j) = λ_{j}^{'} - i$ (Removable border strips and their height, Hooks of a partition correspond bijectively to removable border strips); in terms of parts, with $b_{0} : = ℓ (i, j) = λ_{j}^{'} - i$ , $λ ∖ R_{i, j} = (λ_{1}, \dots, λ_{i - 1}, λ_{i + 1} - 1, \dots, λ_{i + b_{0}} - 1, j - 1, λ_{i + b_{0} + 1}, \dots),$ so that $∣ λ ∖ R_{i, j} ∣ = n - h_{i, j} (λ)$ . Finally $X^{'}$ is obtained from the configuration $A_{X}$ by moving the bead from position $b$ to the gap $g$ .

(iii) In particular $e ∣ h_{i, j} (λ)$ if and only if $b \equiv g (m o d e)$ , that is, if and only if the bead $b$ and the gap $g$ lie on the same runner of the $e$ -abacus; then $g = b - e \cdot (h_{i, j} (λ) / e)$ and the move in (ii) is the slide of the bead $b$ up $h_{i, j} (λ) / e$ rows on its own runner. A hook of length $e$ in row $i$ of $λ$ exists if and only if $h_{i} \geq e$ and $h_{i} - e \notin X$ ; Thus $λ$ has a hook of length $e$ if and only if some bead of $X$ has a gap in the position $e$ labels above it on its own runner.

No choice principle is used.

Facts & Assumptions

Given: An integer $e \geq 2$ , a partition $λ ⊢ n$ with $t : = ℓ (λ)$ parts, and a $β$ -set $X = {h_{1} > \dots > h_{m}}$ for $λ$ , so that $P (X) = λ$ (Beta-sets, the shift, and the e-runner abacus of a partition).

[F1]

Hook lengths: $h_{i, j} (λ) = λ_{i} - j + λ_{j}^{'} - i + 1$ for $(i, j) \in [λ]$ , and for fixed $i$ the map $j \mapsto h_{i, j} (λ)$ is strictly decreasing on $1 \leq j \leq λ_{i}$ , because $h_{i, j} (λ) - h_{i, j + 1} (λ) = 1 + λ_{j}^{'} - λ_{j + 1}^{'} \geq 1$ ; hence the hook lengths in a fixed row are pairwise distinct and $h_{i, j} (λ) \geq 1$ (Hook, arm, leg, and hook length of a box).

[F2]

For a $β$ -set $Y = {y_{1} > \dots > y_{r}}$ one has $P (Y)_{j} = y_{j} - (r - j)$ for $1 \leq j \leq r$ , after which trailing zeros are deleted: indeed there are $y_{j}$ positions $0, 1, \dots, y_{j} - 1$ below the bead $y_{j}$ , of which exactly $r - j$ are occupied by the beads $y_{j + 1}, \dots, y_{r}$ , so $y_{j} - (r - j)$ of them are gaps, and the definition of $P$ takes exactly this difference (Beta-sets, the shift, and the e-runner abacus of a partition).

[F3]

The $β$ -sets of $λ$ are exactly the sets $X_{λ} + l$ , $l \in N_{0}$ , where $X_{λ} = {b_{1} (λ) > \dots > b_{t} (λ)}$ with $b_{i} (λ) = h_{i, 1} (λ) = λ_{i} + t - i$ , and $∣ X_{λ} + l ∣ = t + l$ (Beta-sets, the shift, and the e-runner abacus of a partition).

[F4]

A removable border strip of $λ$ is a nonempty edge-connected set of nodes containing no $2 \times 2$ block whose complement in $[λ]$ is again a Young diagram; its height is its number of occupied rows minus one, and for a strip that is the hook of a node it equals the leg length of that node (Removable border strips and their height, Hooks of a partition correspond bijectively to removable border strips).

Proof

technique · direct

1.1F1F2F3algebra

Pad $λ$ with zeros to length $m$ . By [F3], $h_{i} = λ_{i} + m - i$ . Below $h_{i}$ there are exactly $m - i$ beads, hence $h_{i} - (m - i) = λ_{i}$ gaps. For a node $(i, j)$ put $g_{j} = h_{i} - h_{i, j} (λ) = m + j - 1 - λ_{j}^{'}$ . These values are distinct by [F1], satisfy $0 \leq g_{j} < h_{i}$ , and are gaps: equality $g_{j} = h_{x} = λ_{x} + m - x$ would give $λ_{x} - j + λ_{j}^{'} - x + 1 = 0$ . If $x \leq λ_{j}^{'}$ , both differences are nonnegative; otherwise both are at most $- 1$ , since $λ_{x} < j$ . Neither case gives zero. Thus the $λ_{i}$ numbers $g_{j}$ exhaust the gaps below $h_{i}$ , including the empty set when $λ_{i} = 0$ .

1.2F1F4constructalgebra

Fix $(i, j)$ and put $r = λ_{j}^{'}$ and $g = m + j - 1 - r$ . The rim strip of [F4] has complement with rows $μ_{p} = λ_{p}$ outside $i \leq p \leq r$ , $μ_{p} = λ_{p + 1} - 1$ for $i \leq p < r$ , and $μ_{r} = j - 1$ : in row $p$ the strip occupies the rim nodes with column at least $j$ , whose complement ends at $\max (j - 1, λ_{p + 1} - 1)$ . For $p < r$ one has $λ_{p + 1} \geq j$ , while $λ_{r + 1} < j$ . These formulas give the displayed partition in (ii), with trailing zeros deleted, and its size is $n - h_{i, j} (λ)$ .

2.1F1step 1.1algebra

Consequently, for $h \geq 0$ , $h \in H_{i} (λ)$ exactly when $h \leq h_{i}$ and $h_{i} - h \notin X$ : at $h = 0$ the latter fails because $h_{i} \in X$ , and for $h > h_{i}$ its first inequality fails. For $1 \leq h \leq h_{i}$ this is the gap correspondence in step 1.1. The bead $h_{i}$ determines $i$ , and the distinct row hook lengths determine $j$ , so $(i, j) \mapsto (h_{i}, g_{j})$ is the asserted bijection with length $h_{i} - g_{j}$ .

2.2F2step 1.1step 1.2algebra

The new value $g$ is below $h_{r}$ because $h_{r} - g = λ_{r} - j + 1 \geq 1$ , and, if $r < m$ , above $h_{r + 1}$ because $g - h_{r + 1} = j - λ_{r + 1} \geq 1$ . Hence the decreasing list of $X^{'} = (X ∖ {h_{i}}) \cup {g}$ is $h_{1}, \dots, h_{i - 1}, h_{i + 1}, \dots, h_{r}, g, h_{r + 1}, \dots, h_{m}$ . Applying [F2] gives precisely the parts $μ_{p}$ in step 1.2. Thus $P (X^{'}) = λ ∖ R_{i, j}$ , and the abacus change is the relocation of the bead $h_{i}$ to $g$ .

3.1step 2.1step 2.2algebra∎

The length is $b - g$ , so $e ∣ h_{i, j} (λ)$ exactly when $b \equiv g (m o d e)$ . In that case the bead remains on its runner, moves up $h_{i, j} (λ) / e$ rows, and all runner bead counts are unchanged. Without this divisibility assumption the relocation may change runners. In row $i$ a hook of length $e$ exists exactly when $h_{i} \geq e$ and $h_{i} - e \notin X$ , by step 2.1; existential quantification over $i$ gives the final bead criterion in (iii).

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

The e-core, the e-quotient, and the e-weight

Definition

Let $e \geq 2$ , let $λ ⊢ n$ and let $X = {h_{1} > h_{2} > \dots > h_{m}}$ be a $β$ -set for $λ$ , with first-column hook lengths $b_{i} (λ)$ and $e$ -abacus configuration $A_{X}$ as in Beta-sets, the shift, and the e-runner abacus of a partition; hooks and their lengths are those of Hook, arm, leg, and hook length of a box, and the hook/bead-gap dictionary is Hooks are bead-gap pairs on the abacus.

$e$ -core partition. The partition $λ$ is an $e$ -core when $e \notin H (λ)$ , that is, when no hook of $λ$ has length $e$ ; by Hooks are bead-gap pairs on the abacus(iii) this is the case exactly when no bead of $A_{X}$ has a gap in the position $e$ labels above it on its own runner. The empty partition is an $e$ -core, and so is every $λ$ with $∣ λ ∣ < e$ .

Runner label sets. For $i \in {0, 1, \dots, e - 1}$ let $X_{i}^{(e)} : = {a \in N_{0} : a e + i \in X}$ be the set of row labels of the beads of $A_{X}$ on runner $i$ . The $X_{i}^{(e)}$ are finite subsets of $N_{0}$ and may overlap. The disjoint union $⨆_{i} {i} \times X_{i}^{(e)}$ records each bead with its runner and row; the map $(i, a) \mapsto a e + i$ bijects it with $X$ .

$e$ -core. The top-packed $β$ -set of $X$ is $X^{(e)} : = ⋃_{i = 0}^{e - 1} {a e + i : 0 \leq a \leq ∣ X_{i}^{(e)} ∣ - 1},$ the configuration obtained from $A_{X}$ by sliding every bead up as far as it goes on its own runner, i.e. into the topmost $∣ X_{i}^{(e)} ∣$ positions of runner $i$ . The $e$ -core of $λ$ is $C_{e} (λ) : = P (X^{(e)}) .$

$e$ -quotient and $e$ -weight. Choose the $β$ -set $X$ of $λ$ to be normalised, meaning $∣ X ∣ \equiv 0 (m o d e)$ . Then define the $e$ -quotient of $λ$ to be the $e$ -tuple of partitions $Q_{e} (λ) : = (λ^{(0)}, λ^{(1)}, \dots, λ^{(e - 1)}), λ^{(i)} : = P (X_{i}^{(e)}),$ each runner read as a $1$ -abacus, and the $e$ -weight of $λ$ to be $w_{e} (λ) : = ∣ Q_{e} (λ) ∣ : = \sum_{i = 0}^{e - 1} ∣ λ^{(i)} ∣ .$ A normalised $β$ -set exists: by Beta-sets, the shift, and the e-runner abacus of a partition the $β$ -sets of $λ$ are the sets $X_{λ} + l$ , $l \in N_{0}$ , and $∣ X_{λ} + l ∣ = ∣ X_{λ} ∣ + l$ , so among the $e$ consecutive shifts $l, \dots, l + e - 1$ exactly one has size divisible by $e$ .

No choice principle is used.

Remarks

Well-definedness. Under the shift $X \mapsto X + 1$ , the runner sets satisfy $(X + 1)_{0}^{(e)} = (X_{e - 1}^{(e)}) + 1$ and $(X + 1)_{i}^{(e)} = X_{i - 1}^{(e)}$ for $1 \leq i < e$ , where $+ 1$ is the beta-set shift, including the new bead at row $0$ . Thus the new counts are $(c_{e - 1} + 1, c_{0}, \dots, c_{e - 2})$ , not just a permutation of the old counts. Top-packing commutes with this shift: $(X + 1)^{(e)} = (X^{(e)}) + 1$ , so $P$ is unchanged. A shift by $e$ applies a beta-set shift by $1$ to every runner, preserving each component partition. Hence any two normalised beta-sets give the same ordered quotient and weight. These identities and the identification of the core with terminal rim $e$ -hook removal are proved in The e-core is independent of the rim hook removal order ↗.
$e$ -cores are $e$ -regular and $e$ -restricted. If a positive part $a$ occurs at least $e$ times, put $r = λ_{a}^{'}$ . The bottom $e$ rows of length $a$ are $r - e + 1, \dots, r$ , and the node $(r - e + 1, a)$ has hook length $a - a + r - (r - e + 1) + 1 = e$ . Thus an $e$ -core is $e$ -regular. Conjugation preserves hook lengths, since it exchanges arm and leg; applying this argument to $λ^{'}$ and using e-regular and e-restricted partitions gives $e$ -restrictedness.
Normalisation and the quotient's ordering. Shifting by $1$ rotates the component partitions cyclically. Shifting by $e$ increases every runner count by $1$ and replaces each row-label set $Y$ by $Y + 1 = {a + 1 : a \in Y} \cup {0}$ ; it preserves $P (Y)$ , rather than $Y$ itself. The condition $e ∣ ∣ X ∣$ therefore fixes the ordered quotient, as in Lemma 4.9 and its normalising remark in the lecture notes.

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

Virtual Weyl combinations detect q-Schur block separation

Statement

Let $F$ be a field, $q \in F^{\times}$ , $d, n \geq 1$ , and let $S = S (d, n)$ be the $q$ -Schur algebra over $F$ with Weyl modules $W^{λ}$ for $λ \in Λ^{+} (d, n)$ (The type-A q-Schur algebra and its Weyl modules). Let $M$ be a finite dimensional right $S$ -module all of whose composition factors lie in a single block $B$ of $S$ (Blocks of a finite-dimensional algebra via primitive central idempotents), and let integers $a_{λ}$ , $λ \in Λ^{+} (d, n)$ , satisfy $[M] = \sum_{λ \in Λ^{+} (d, n)} a_{λ} [W^{λ}]$ in the Grothendieck group $G_{0} (S -mod)$ (Grothendieck group of an essentially small abelian category). Then $W^{λ}$ lies in $B$ whenever $a_{λ} \neq 0$ . In particular no nonzero virtual combination of Weyl classes all of whose terms lie outside $B$ can equal the class of a module in $B$ .

No choice principle is used.

Facts & Assumptions

Given: A field $F$ , $q \in F^{\times}$ , $d, n \geq 1$ , the algebra $S = S (d, n)$ with Weyl modules $W^{λ}$ , $λ \in Λ^{+} (d, n)$ , a finite dimensional right $S$ -module $M$ whose composition factors all lie in one block $B$ , and integers $a_{λ}$ with $[M] = \sum_{λ} a_{λ} [W^{λ}]$ in $G_{0} (S -mod)$ (The type-A q-Schur algebra and its Weyl modules, Grothendieck group of an essentially small abelian category).

[F1]

Blocks: $1 = \sum_{i} ϵ_{i}$ is the unique decomposition of the identity of $S$ into pairwise orthogonal primitive central idempotents, the blocks are the two-sided ideals $B_{i} = S ϵ_{i} = ϵ_{i} S$ , and for every right $S$ -module $N$ one has $N = ⨁_{i} N ϵ_{i}$ ; a simple module $L$ lies in exactly one block and satisfies $L ϵ_{i} = L$ for that $i$ (Blocks of a finite-dimensional algebra via primitive central idempotents). We write $ϵ_{B}$ for the idempotent of the block $B$ of the statement.

[F2]

Each Weyl module has nonzero simple head $L^{λ}$ (The type-A q-Schur algebra and its Weyl modules).

[F3]

The proof of Semistandard weight dimensions detect virtual Weyl classes, steps 1.1 and 3.1, gives additive weight dimensions and the weight matrix $K_{λ ν} = \dim (W^{λ} 1_{ν})$ . This matrix, restricted to partitions with at most $d$ parts, is unitriangular and hence invertible by The Kostka change of basis is dominance-unitriangular. Applying these dimensions to a zero virtual Weyl combination forces every coefficient to vanish. This proves the linear independence used here; no Weyl filtration of an arbitrary module is assumed.

[F4]

An exact functor between abelian categories induces a homomorphism of Grothendieck groups, and the finite-dimensional $S$ -modules form an abelian category in which $G_{0}$ is computed by composition multiplicities (Grothendieck group of an essentially small abelian category).

Proof

technique · direct

1.1F1F4

Multiplication by the central idempotent $ϵ_{B}$ is the projection of every right $S$ -module onto its $B$ -summand: $N = ⨁_{i} N ϵ_{i}$ by [F1], so $N ϵ_{B}$ is that direct summand and is a submodule, and $N ϵ_{B} = N$ exactly when all composition factors of $N$ lie in $B$ . The assignment $N \mapsto N ϵ_{B}$ is exact, because it is the projection onto a direct summand of the identity functor: for $f : N \to N^{'}$ one has $f (N ϵ_{B}) = f (N) ϵ_{B}$ , $\ker (f ∣_{N ϵ_{B}}) = (\ker f) ϵ_{B}$ and $im (f ∣_{N ϵ_{B}}) = (im f) ϵ_{B}$ , so it preserves kernels and images.

1.2F1F2algebra

A nonzero finite-dimensional module has a simple quotient: choose a proper submodule of maximal vector-space dimension. If a Weyl module had two nonzero summands, each would have a nonzero head, so its head would not be simple, contrary to [F2]. Thus it is indecomposable. Its block decomposition [F1] has precisely one nonzero summand, the block containing its head $L^{λ}$ ; hence $W^{λ} ϵ_{B}$ is either $W^{λ}$ or $0$ .

2.1F4step 1.1step 1.2algebra

Apply the exact projection of step 1.1 to the class identity of the hypothesis. Since $M ϵ_{B} = M$ by the hypothesis that all composition factors of $M$ lie in $B$ , and since $W^{λ} ϵ_{B}$ is $W^{λ}$ or $0$ by step 1.2, the induced identity in $G_{0}$ reads $[M] = \sum_{λ : W^{λ} \in B} a_{λ} [W^{λ}] .$ Subtracting it from the given identity $[M] = \sum_{λ} a_{λ} [W^{λ}]$ yields $\sum_{λ : W^{λ} \notin B} a_{λ} [W^{λ}] = 0$ in $G_{0} (S -mod)$ .

3.1F3step 2.1∎

By the linear independence of the Weyl classes [F3], all coefficients in the vanishing combination of step 2.1 are zero: $a_{λ} = 0$ for every $λ$ with $W^{λ} \notin B$ , which is the contrapositive of the assertion that $a_{λ} \neq 0$ implies $W^{λ} \in B$ .

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

The e-core is independent of the rim hook removal order

Statement

Let $e \geq 2$ and let $λ ⊢ n$ with $e$ -abacus configurations as in Beta-sets, the shift, and the e-runner abacus of a partition and the top-packed set $X^{(e)}$ of The e-core, the e-quotient, and the e-weight. For every $β$ -set $X$ of $λ$ the partition $P (X^{(e)})$ is the same, so the $e$ -core $C_{e} (λ) = P (X^{(e)})$ is a well-defined $e$ -core partition, and it is exactly the terminal partition reached by successively removing removable border strips of size $e$ from $[λ]$ until none is left (Removable border strips and their height, Hooks of a partition correspond bijectively to removable border strips). In particular every maximal sequence of such removals gives the same terminal partition $C_{e} (λ)$ , each removed strip has size exactly $e$ , and the terminal partition contains no hook of length $e$ . Moreover any two normalised beta-sets ( $e ∣ ∣ X ∣$ ) give the same ordered quotient $Q_{e} (λ)$ and weight.

No choice principle is used.

Facts & Assumptions

Given: An integer $e \geq 2$ , a partition $λ ⊢ n$ , and a $β$ -set $X$ of $λ$ with runner label sets $X_{i}^{(e)}$ , counts $c_{i} : = ∣ X_{i}^{(e)} ∣$ and top-packed set $X^{(e)}$ as in The e-core, the e-quotient, and the e-weight.

[F1]

Hooks correspond to bead-gap pairs $(b, g)$ with length $b - g$ ; deleting the corresponding rim strip replaces $b$ by $g$ . If the length is divisible by $e$ , the relocation stays on the same runner (Hooks are bead-gap pairs on the abacus).

[F2]

The top-packed set has runner $i$ occupied in rows $0, \dots, c_{i} - 1$ , and $C_{e} (λ) = P (X^{(e)})$ . The ordered quotient is computed from the partitions of the runner row-label sets of a normalised beta-set (The e-core, the e-quotient, and the e-weight).

[F3]

Every beta-set of $λ$ is $X_{λ} + l$ for a unique $l \geq 0$ , and $P (X + l) = P (X)$ . Thus, after interchanging two beta-sets if necessary, the larger one is a shift of the smaller one (Beta-sets, the shift, and the e-runner abacus of a partition).

[F4]

The hook/rim-strip correspondence is a bijection preserving size (Removable border strips and their height, Hooks of a partition correspond bijectively to removable border strips).

Proof

technique · direct

1.1F1F2

Fix a $β$ -set $X$ of a partition $μ$ and write its elements as $h_{1} > \dots > h_{m}$ . By [F1] the hooks of length $e$ of $μ$ correspond bijectively to the beads $b = h_{i}$ with $h_{i} \geq e$ and $h_{i} - e \notin X$ , the correspondence sending such a bead to the rim strip $R_{i, j} (μ)$ attached to the node with $h_{i, j} (μ) = e$ , and removal of that strip replaces $X$ by $X^{'} = (X ∖ {h_{i}}) \cup {h_{i} - e}$ , a $β$ -set for $μ ∖ R_{i, j} (μ)$ of the same size $m$ . In particular $μ$ has no hook of length $e$ exactly when no bead of $X$ has a gap in the position $e$ labels above it on its own runner.

1.2F2F3algebra

Compute the shift directly. For $Y = X + 1$ , its runner sets are $Y_{0}^{(e)} = (X_{e - 1}^{(e)}) + 1$ and $Y_{i}^{(e)} = X_{i - 1}^{(e)}$ for $1 \leq i < e$ : the last runner wraps to runner $0$ one row lower, and the shift adds the bead at $0$ . Its counts are $(c_{e - 1} + 1, c_{0}, \dots, c_{e - 2})$ . Packing with these counts gives $Y^{(e)} = (X^{(e)}) + 1$ , so $P (Y^{(e)}) = P (X^{(e)})$ by [F3]. Iteration proves core independence for all beta-sets of $λ$ . After $e$ shifts, every runner row-label set has undergone one beta-set shift by $1$ , so its partition is unchanged. Normalised beta-sets differ by a multiple of $e$ , proving independence of the ordered quotient and its total size.

2.1F2step 1.1

Call a bead of $X$ movable when the position $e$ labels above it on its runner is a gap, so that by step 1.1 the movable beads are exactly the hooks of length $e$ of $μ$ ; moving a bead up one row replaces $X$ by a $β$ -set of $μ$ minus that hook, and the counts $∣ X_{i}^{(e)} ∣$ of beads on each runner are unchanged. A configuration is top-packed precisely when it has no movable bead: a top-packed runner carries beads in the rows $0, 1, \dots, c_{i} - 1$ , so the row above a bead is a bead again (and row $0$ has no row above it), while if some runner is not top-packed there are a gap in row $g$ and a bead in a row $a > g$ on that runner; choosing $a$ minimal among the bead rows larger than $g$ , the row $a - 1$ is a gap, so the bead in row $a$ is movable.

3.1F2step 1.1step 2.1algebra

Every sequence of moves of steps 1.1 and 2.1 is finite: the quantity $Φ : = \sum_{i} \sum_{a \in X_{i}^{(e)}} a$ is the total of the row labels of all beads, and each move replaces a bead's row label $a$ by $a - 1$ , so $Φ$ decreases by exactly $1$ and is a nonnegative integer. A sequence is maximal exactly when it stops at a configuration with no movable bead, that is, by step 2.1, exactly when it stops at the top-packed configuration of the counts $(c_{i})$ ; and that configuration is unique, since a runner with $c_{i}$ beads has exactly one top-packed state, the rows $0, \dots, c_{i} - 1$ . Hence every maximal sequence of moves starting from $X$ ends at the same configuration, namely $X^{(e)}$ , and therefore at the same terminal partition $P (X^{(e)})$ .

4.1F2F4step 1.1step 2.1step 3.1step 1.2∎

By step 1.2 the partition $P (X^{(e)})$ does not depend on the chosen $β$ -set $X$ , so $C_{e} (λ) : = P (X^{(e)})$ is well-defined; by step 3.1 it is the common terminal partition of all maximal sequences of e-hook removals, and by step 2.1 a terminal configuration is top-packed, hence carries no movable bead, hence its partition has no hook of length $e$ by step 1.1; thus $C_{e} (λ)$ is an $e$ -core. Every strip removed along such a sequence corresponds to a node hook of length exactly $e$ and so has size $e$ by [F4]. This proves all the asserted properties.

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

The e-quotient controls the hooks of e-divisible length

Statement

Let $e \geq 2$ and let $λ ⊢ n$ with $e$ -quotient $Q_{e} (λ) = (λ^{(0)}, \dots, λ^{(e - 1)})$ and $e$ -core $C_{e} (λ)$ (The e-core, the e-quotient, and the e-weight). Then there is a bijection $f$ from the set of hooks of $λ$ of $e$ -divisible length onto the set of hooks of the multipartition $Q_{e} (λ)$ , that is, onto $⨆_{i = 0}^{e - 1} {hooks of λ^{(i)}}$ , such that $∣ H ∣ = e ∣ f (H) ∣ and Q_{e} (λ ∖ H) = Q_{e} (λ) ∖ f (H),$ where the right-hand side means that the hook $f (H)$ is removed from the single component of $Q_{e} (λ)$ that contains it. On the $e$ -abacus, $f$ is the rescaling $b = b^{'} e + i$ , $g = g^{'} e + i$ of a bead-gap pair $(b, g)$ lying on a common runner $i$ , followed by the bead-gap dictionary of the runner (Hooks are bead-gap pairs on the abacus).

Here and in the quotient components, removing a hook means deleting its corresponding rim border strip as in Hooks are bead-gap pairs on the abacus(ii), rather than its arm-and-leg node set.

No choice principle is used.

Facts & Assumptions

Given: An integer $e \geq 2$ , a partition $λ ⊢ n$ , a normalised $β$ -set $X = {h_{1} > \dots > h_{m}}$ of $λ$ , so that $e ∣ m = ∣ X ∣$ , with runner label sets $X_{i}^{(e)}$ , top-packed set $X^{(e)}$ and quotient components $λ^{(i)} = P (X_{i}^{(e)})$ as in The e-core, the e-quotient, and the e-weight.

[F1]

For a $β$ -set $Y$ of a partition $μ$ the hooks of $μ$ correspond bijectively to the bead-gap pairs $(b, g)$ of $Y$ , with $b \in Y$ , $g \notin Y$ and $0 \leq g < b$ , the hook of $(b, g)$ having length $b - g$ ; the hook has $e$ -divisible length if and only if $b \equiv g (m o d e)$ ; and removing the hook of $(b, g)$ replaces $Y$ by $(Y ∖ {b}) \cup {g}$ , a $β$ -set for the partition $μ$ with that hook removed (Hooks are bead-gap pairs on the abacus).

[F2]

For $i \in {0, \dots, e - 1}$ the runner label set $X_{i}^{(e)}$ is a $β$ -set for the partition $λ^{(i)} = P (X_{i}^{(e)})$ , and position $a e + i$ of the abacus lies on runner $i$ at row $a$ , so that the beads of $X_{i}^{(e)}$ are exactly the beads of $X$ on runner $i$ (The e-core, the e-quotient, and the e-weight, Beta-sets, the shift, and the e-runner abacus of a partition).

[F3]

The chosen $β$ -set is normalised, $e ∣ m$ ; the removal of a hook does not change the number of beads, so the new $β$ -set remains normalised (The e-core, the e-quotient, and the e-weight).

Proof

technique · direct

1.1F1F2

Write a hook $H = H_{i, j} (λ)$ of $e$ -divisible length in its bead-gap form $(b, g)$ with $b \in X$ , $g \notin X$ , $0 \leq g < b$ and $h : = ∣ H ∣ = b - g$ . By $e ∣ h$ we have $b \equiv g (m o d e)$ ; write the common residue as $s \in {0, \dots, e - 1}$ and $b = b^{'} e + s, g = g^{'} e + s, b^{'} = \frac{b - s}{e} > g^{'} = \frac{g - s}{e} \geq 0.$ The bead $b$ is a bead of $X$ on runner $s$ and the gap $g$ is a gap of $X$ on runner $s$ .

2.1F1F2step 1.1

Applying step 1.1 conversely, a bead $b^{'} \in X_{s}^{(e)}$ and a gap $g^{'} \notin X_{s}^{(e)}$ with $0 \leq g^{'} < b^{'}$ give the bead-gap pair $(b, g) = (b^{'} e + s, g^{'} e + s)$ of $X$ with $e ∣ (b - g)$ and $b - g = e (b^{'} - g^{'})$ . Hence $f (H) : =$ the hook of $λ^{(s)}$ attached to the bead-gap pair $(b^{'}, g^{'})$ of the $β$ -set $X_{s}^{(e)}$ is well defined, and $f$ is bijective onto the hooks of $⨆_{i} λ^{(i)}$ because both sides are in bijection with the bead-gap pairs of $X$ whose two positions share a runner, respectively with the union over $s$ of the bead-gap pairs of $X_{s}^{(e)}$ ; the length identity is $∣ H ∣ = b - g = e (b^{'} - g^{'}) = e ∣ f (H) ∣$ .

3.1F1F2F3step 1.1step 2.1

For the removal identity, let $X^{'} = (X ∖ {b}) \cup {g}$ be the $β$ -set of $λ ∖ H$ from [F1], again of size $m$ and hence normalised by [F3]. Because $b$ and $g$ lie on runner $s$ , the beads of $X^{'}$ on a runner $t \neq s$ are exactly the beads of $X$ on runner $t$ , and the beads on runner $s$ have row labels $(X_{s}^{(e)} ∖ {b^{'}}) \cup {g^{'}}$ . Consequently, computing the quotient from the normalised sets by [F2], $λ^{' (t)} = P ((X^{'})_{t}^{(e)}) = λ^{(t)} (t \neq s), λ^{' (s)} = P ((X_{s}^{(e)} ∖ {b^{'}}) \cup {g^{'}}),$ and the last partition is $λ^{(s)}$ with its hook $f (H)$ removed, by [F1] applied to the $β$ -set $X_{s}^{(e)}$ of $λ^{(s)}$ . This is exactly the assertion $Q_{e} (λ ∖ H) = Q_{e} (λ) ∖ f (H)$ .

4.1step 1.1step 2.1step 3.1∎

Steps 2.1 and 3.1 together exhibit the required bijection $f$ with the length and removal identities, and the description of $f$ on the abacus is the rescaling of step 1.1 followed by the bead-gap dictionary of the runner in step 2.1.

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

The core-quotient size formula defines the weight

Statement

Let $e \geq 2$ and let $λ ⊢ n$ with $e$ -core $C_{e} (λ)$ and $e$ -weight $w_{e} (λ) = ∣ Q_{e} (λ) ∣$ (The e-core, the e-quotient, and the e-weight). Then $n = ∣ λ ∣ = ∣ C_{e} (λ) ∣ + e w_{e} (λ),$ so $w_{e} (λ) = (n - ∣ C_{e} (λ) ∣) / e \in N_{0}$ . Equivalently every sequence of removals of removable border strips of size $e$ from $λ$ down to its $e$ -core (The e-core is independent of the rim hook removal order) has exactly $w_{e} (λ)$ steps, and $w_{e} (λ) = ∣ Q_{e} (λ) ∣$ equals both the number of hooks of the multipartition $Q_{e} (λ)$ and, by the bijection of The e-quotient controls the hooks of e-divisible length, the number of hooks of $λ$ of length divisible by $e$ .

Throughout the proof, $λ ∖ H$ means deletion of the rim strip corresponding to the node hook $H$ , as in The e-quotient controls the hooks of e-divisible length.

No choice principle is used.

Facts & Assumptions

Given: An integer $e \geq 2$ and a partition $λ ⊢ n$ with $e$ -quotient $Q_{e} (λ) = (λ^{(0)}, \dots, λ^{(e - 1)})$ , $e$ -core $C_{e} (λ)$ and $e$ -weight $w_{e} (λ) = \sum_{i} ∣ λ^{(i)} ∣$ (The e-core, the e-quotient, and the e-weight).

[F1]

$C_{e} (λ)$ is well defined and is the terminal partition of every maximal sequence of removals of removable border strips of size $e$ from $λ$ ; each removed strip has size exactly $e$ and the terminal partition contains no hook of length $e$ (The e-core is independent of the rim hook removal order).

[F2]

There is a bijection $f$ from the hooks of $λ$ of $e$ -divisible length onto the hooks of the multipartition $Q_{e} (λ)$ with $∣ H ∣ = e ∣ f (H) ∣$ and $Q_{e} (λ ∖ H) = Q_{e} (λ) ∖ f (H)$ , where the latter removes the hook $f (H)$ from its component (The e-quotient controls the hooks of e-divisible length).

[F3]

For a partition $μ$ , the number of hooks of $μ$ equals $∣ μ ∣$ , since each box is the anchor of exactly one hook of $[μ]$ (Hook, arm, leg, and hook length of a box); a nonempty partition has a removable box, that is, a hook of size $1$ (Removable and addable nodes); and a hook of the multipartition $Q_{e} (λ)$ has the size of the corresponding hook of the component $λ^{(i)}$ (The e-core, the e-quotient, and the e-weight, The e-quotient controls the hooks of e-divisible length).

Proof

technique · induction on $w_e(\lambda)$

1.1F2F3

We first record the counting consequences of the quotient theorem. A hook of $λ$ has $e$ -divisible length exactly when it is the preimage under the bijection $f$ of [F2] of a hook of $Q_{e} (λ)$ , and the number of hooks of a partition equals its size by [F3]; hence the number of $e$ -divisible hooks of $λ$ equals the total size $w_{e} (λ) = \sum_{i} ∣ λ^{(i)} ∣$ of $Q_{e} (λ)$ . In particular $Q_{e} (λ)$ has a hook if and only if $w_{e} (λ) > 0$ , and then some hook has size $1$ by [F3], so $λ$ has a hook of length $e$ .

2.1F1basestep 1.1

Base case: if $w_{e} (λ) = 0$ then $Q_{e} (λ)$ has no hook by step 1.1, so $λ$ has no hook of $e$ -divisible length, hence no hook of length $e$ ; by [F1] the terminal partition of the empty removal sequence is $λ$ itself, that is, $C_{e} (λ) = λ$ , and the removal sequence has $0 = w_{e} (λ)$ steps.

2.2F1F2F3step 1.1

Induction step: let $w_{e} (λ) > 0$ . By step 1.1 there is a hook $H$ of $λ$ of length exactly $e$ , a removable border strip of size $e$ by [F1]. By the removal identity of [F2], $Q_{e} (λ ∖ H) = Q_{e} (λ) ∖ f (H),$ and since $f (H)$ is a hook of a component of $Q_{e} (λ)$ of size $∣ H ∣ / e = 1$ , the right-hand side is the multipartition obtained by deleting one box, so by [F3] its size is $w_{e} (λ) - 1$ . Moreover $C_{e} (λ ∖ H) = C_{e} (λ)$ : a maximal sequence from $λ ∖ H$ is the tail of a maximal sequence from $λ$ beginning with $H$ , and all maximal sequences from $λ$ end at $C_{e} (λ)$ by [F1].

3.1F1IHstep 2.1step 2.2algebra

By step 2.2 we may apply the induction hypothesis to $λ ∖ H$ , whose weight is $w_{e} (λ) - 1 < w_{e} (λ)$ : every maximal sequence of $e$ -strip removals from $λ ∖ H$ has $w_{e} (λ) - 1$ steps and $∣ λ ∣ - e = ∣ λ ∖ H ∣ = ∣ C_{e} (λ ∖ H) ∣ + e (w_{e} (λ) - 1) = ∣ C_{e} (λ) ∣ + e w_{e} (λ) - e,$ so $∣ λ ∣ = ∣ C_{e} (λ) ∣ + e w_{e} (λ)$ ; and appending the single step $H$ shows that every maximal sequence from $λ$ has exactly $w_{e} (λ)$ steps, because $H$ was an arbitrary possible first $e$ -hook removal and step 2.2 applies to every such first step.

4.1step 1.1step 2.1step 3.1discharge-induction: step 2.2∎

The base case of step 2.1 and the induction step of steps 2.2 and 3.1 establish the size formula and the step count for every partition of finite weight, and the counting statement is step 1.1. This is all that was asserted.

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

q-Schur Weyl modules are in the same block iff their labels have the same e-core

Statement

Let $F$ be a field and $q \in F^{\times}$ with quantum characteristic $e$ (The quantum characteristic of an invertible Hecke parameter, The $q$ -integer, $q$ -factorial and $q$ -multinomial coefficients), let $n \geq 0$ and $d \geq n$ , and let $W_{F}^{λ}$ be the Weyl modules of $S_{F} (d, n)$ (The type-A q-Schur algebra and its Weyl modules). Then for partitions $λ, μ ⊢ n$ with at most $d$ parts, $W_{F}^{λ}$ and $W_{F}^{μ}$ lie in the same block of $S_{F} (d, n)$ if and only if $λ$ and $μ$ have the same $e$ -core (The e-core, the e-quotient, and the e-weight). Consequently, for finite $e$ , the blocks of $S_{F} (d, n)$ are indexed by the $e$ -cores $γ$ with $∣ γ ∣ \leq n$ and $∣ γ ∣ \equiv n (m o d e)$ , and the block of $W_{F}^{λ}$ is determined by $C_{e} (λ)$ . For $e = \infty$ we set $C_{\infty} (λ) : = λ$ ; the algebra is semisimple and its blocks are indexed by the partitions of $n$ .

No choice principle is used.

Facts & Assumptions

Given: A field $F$ , parameter $q$ , quantum characteristic $e$ , integer $d \geq n$ , and the Weyl modules in the statement. Write $S = S_{F} (d, n)$ and $W^{λ} = W_{F}^{λ}$ .

[F1]

The semistandard cellular basis has cells $W^{λ}$ , nonzero absolutely simple heads $L^{λ}$ , and dominance-unitriangular decomposition multiplicities $d_{λ μ}$ (The type-A q-Schur algebra and its Weyl modules). The cellular anti-involution induces contragredient duality fixing the simple isomorphism classes, through the nondegenerate cell form on each simple head (The cell form is associative and its radical is a submodule, Nonzero cell heads classify the simple modules of a cellular algebra).

[F2]

The balanced hook-ratio Jantzen formula and alternating beta symbols (The q-Schaper virtual radical-layer formula, Virtual Weyl and Specht symbols attached to beta sequences) use conjugate beta numbers $β_{j} = λ_{j}^{'} + N - j$ , with $N$ sufficiently large, and read $\sum_{r > 0} [W^{λ} (r)] = \sum_{(a, b), (a, c) \in [λ], b < c} (v (h_{a b}^{λ}) - v (h_{a c}^{λ})) [W^{β + h_{a c}^{λ} ε_{b} - h_{a c}^{λ} ε_{c}}],$ where $W^{λ} (r)$ are filtration submodules, $W^{λ} (1) = rad W^{λ}$ , and the right hand symbols are zero on repeated or negative entries and otherwise have the sign of sorting the beta sequence. The suppliers must carry this balanced form; the earlier displayed unbalanced manuscript indices do not preserve degree.

[F3]

Core abaci are obtained by packing beads upwards on each runner, so their shapes depend only on runner bead counts; rim-hook removal is a bead-to-gap move (The e-core, the e-quotient, and the e-weight, The e-core is independent of the rim hook removal order, Beta-sets, the shift, and the e-runner abacus of a partition). Conjugation reverses dominance (Dominance order on partitions). Transposing every strip in a maximal $e$ -hook removal sequence gives such a sequence for the conjugate partition; hook lengths are preserved by transposition, so its terminal core is the conjugate core. Hence $C_{e} (λ^{'}) = C_{e} (λ)^{'}$ .

[F4]

Primitive central idempotents decompose the identity, and every finite-dimensional module splits into their block summands (Blocks of a finite-dimensional algebra via primitive central idempotents). Weyl classes are linearly independent in the Grothendieck group, so a nonzero accumulated Weyl coefficient in the class of a module supported in one block forces that Weyl module into the block (Virtual Weyl combinations detect q-Schur block separation, Semistandard weight dimensions detect virtual Weyl classes).

[F5]

The radical of a finite-dimensional algebra is nilpotent and its quotient semisimple (For a finite-dimensional algebra, the Jacobson radical is nilpotent and the quotient by it is semisimple); the Wedderburn decomposition is a product of matrix rings over division rings (Wedderburn–Artin theorem for semisimple rings). Cellular ideal layers are the tensor products of the contragredient left cell and the right cell (Cellular bases and cell modules).

Proof

technique · direct

1.1givenF2algebra

Lift and valuation. Use $R = F [X]_{(X - q)}$ with $\hat{q} = X$ and uniformizer $X - q$ . The element $X$ is a unit since $q \neq 0$ , and every polynomial $[m]_{X} = 1 + X + \dots + X^{m - 1}$ is nonzero, so the generic factorial is nonzero. For $n = 0$ , $S_{F} (d, 0) = F$ , with its one empty-label module, so all assertions hold directly; hence assume $n > 0$ . Its valuation $v (m)$ is positive exactly when $[m]_{q} = 0$ , equivalently $e ∣ m$ for finite $e$ ; for $e = \infty$ all valuations vanish. In characteristic $p$ the positive valuation may exceed1, which causes no difficulty below.

1.2F1F2F4

Block support and Jantzen edges. A finite-dimensional module with simple head is indecomposable: every nonzero summand has nonzero head, so two summands would make the head nonsimple. Thus $W^{λ}$ has just one nonzero block summand, and every Jantzen submodule is supported in that block. By [F4], any nonzero accumulated coefficient $J_{λ μ}$ of $[W^{μ}]$ in [F2] puts $W^{λ}, W^{μ}$ in the same block. This uses the sum of submodule classes, without identifying it with the radical class.

2.1F2F3step 1.1algebra

Balanced terms preserve the core and descend in dominance. Put $s = h_{a b}$ and $t = h_{a c}$ , so $s - t = β_{b} - β_{c}$ . If a hook coefficient is nonzero, at least one of $s, t$ is divisible by $e$ . If $e ∣ t$ , the shifted bead residues are unchanged. Otherwise $e ∣ s$ , so $t \equiv β_{c} - β_{b} (m o d e)$ and the two shifted residues exchange. Thus every surviving symbol has the same runner counts and core as $λ$ . Moreover replacing the two conjugate beta entries $u > v$ by $u + t, v - t$ strictly increases conjugate majorization; the sorted target therefore strictly decreases original dominance. Accumulating equal target labels preserves both conclusions.

2.2F2step 1.1algebra

No outgoing edge implies a binary row condition. Suppose every accumulated outgoing coefficient from $μ$ is zero. For row $i$ , put $g_{i} = N + i - 1 - μ_{i}$ , a gap in its conjugate beta set; its row hooks are exactly $β_{j} - g_{i} > 0$ . If that row contained a divisible hook $s$ and a nondivisible hook $t$ , repeatedly replace $t$ by $s + t$ while their sum is another row hook. This finite climb ends with $s + t$ absent from the hook set. The balanced term for the nodes of hook lengths $s, t$ then has lower shifted bead $g_{i}$ , which is a gap, and upper shifted bead $g_{i} + s + t$ , which is absent from the beta set. Its symbol survives and its coefficient is $\pm v (s) \neq 0$ . There is no cancellation: any surviving balanced term replaces two lost values by a new value larger than both and a new value smaller than both, so those four values determine the pair and its row gap uniquely; the gaps $g_{i}$ strictly increase with $i$ . This contradicts the zero accumulated coefficient. Therefore each row has either all hook lengths divisible by $e$ or none; call this the binary row condition.

3.1F1F2step 2.1algebra

Composition-factor cores. An off-head simple occurs in $W^{λ}$ exactly when it occurs in its radical, equivalently in the positive Jantzen sum: the first submodule is the radical and every later one is contained in it. Its nonzero multiplicity in that sum implies it occurs in some $W^{μ}$ with $J_{λ μ} \neq 0$ , since otherwise its coefficient in the sum would be zero. Step 2.1 gives the same core and strict dominance descent. Induction on the finite dominance poset proves that every factor $L^{ν}$ of $W^{λ}$ has $C_{e} (ν) = C_{e} (λ)$ . No identity equating the radical class with the positive sum is asserted.

3.2F3step 2.2algebra

Vertical core removal under the binary row condition. A removable $e$ -hook has hook length $e$ at its defining node. Every hook in that row is then divisible, so its last-column hook, a positive multiple of $e$ no larger than $e$ , equals $e$ . Hook lengths strictly decrease across the row, forcing the defining node to be at the row end: the strip is vertical on $e$ equal-length rows $i, \dots, i + e - 1$ in column $j$ . Removing it decreases that column height by $e$ . In unaffected rows hook divisibility is unchanged. For an affected row $i + t$ , hooks in the retained columns were all congruent to $- t$ modulo $e$ , because row $i$ had all hooks divisible and those rows had equal length. Their new hook lengths are all congruent to $- t - 1$ , so the binary row condition persists. Induction shows every core-removal strip is vertical.

4.1F1F4F5step 3.1algebra

The Cartan graph locally. Choose a complete orthogonal primitive idempotent system of $S$ , with one representative $f_{μ}$ for each simple label and repeated representatives as necessary. By [F5] and the absolute simplicity in [F1], the semisimple quotient is a product of full matrix rings over $F$ . Lift its matrix diagonal idempotents: the radical is nilpotent and the iteration $u \mapsto u + (1 - 2 u) (u^{2} - u)$ changes an error $a = u^{2} - u$ to $- 3 a^{2} + 4 a^{3}$ , so finite iteration lifts an idempotent; successively working in complementary corners lifts an orthogonal system. Absolute simplicity gives $\dim L^{λ} f_{μ} = δ_{λ μ}$ , hence exact multiplication by $f_{μ}$ gives $\dim W^{λ} f_{μ} = d_{λ μ}$ . The successive cellular ideal bimodules are the contragredient cell module tensor the right cell module. Apply the exact projection $f_{μ} (-) f_{ν}$ and add their dimensions; [F1]'s duality fixes the simple labels, giving $\dim f_{μ} S f_{ν} = \sum_{λ} d_{λ μ} d_{λ ν}$ . This is the Cartan entry of the primitive projective $f_{μ} S$ . A disconnected graph on the full primitive idempotent system, with edges whenever either mixed corner is nonzero, yields central idempotents by summing each component: all mixed corners vanish in both directions. Conversely a central block idempotent forbids mixed corners. Thus simple labels in one block are joined by pairs sharing a Weyl factor, and step 3.1 proves that labels in the same block have the same core.

4.2F3step 3.2constructalgebra

The minimum label and the bridge shapes. Assume $e < \infty$ . Fix core $γ$ and weight $w$ , with $l = ℓ (γ)$ , and set $ρ = (γ_{1}, \dots, γ_{l}, 1^{e w})$ . It has core $γ$ by removing its bottom vertical $e$ -strips, and is dominance-minimal in the class: every same-core partition contains $γ$ as a diagram, and appending all additional boxes as rows of size1 minimizes each dominance partial sum. Let $μ \neq ρ$ satisfy the binary row condition. Choose the first row $i$ with divisible hooks and put $j = μ_{i}$ , $h = h_{i j}^{μ} = μ_{j}^{'} - i + 1$ , a positive multiple of $e$ . If $j = 1$ , every divisible hook is in column1, so step 3.2 would give $μ = ρ$ ; thus $j > 1$ . Remove the vertical $h$ -strip, obtaining $σ$ , then add $h$ boxes horizontally to its first row to get $ν$ , and vertically to its first column to get $η$ . All have the same core, since $h$ is a multiple of $e$ ; $ν$ dominates $μ, η$ , and $η$ is strictly below $μ$ because the removed nodes are moved into new bottom rows.

5.1F2step 1.1step 2.2step 4.2algebra

The bridge coefficients are nonzero. Write $l = ℓ (μ) = ℓ (σ)$ . If $i > 1$ , then $σ_{1} = μ_{1}$ and the row1 hook pair in $ν$ at columns $j, μ_{1} + 1$ has lengths $A = h_{1 j}^{μ}$ and $h$ : the first is nondivisible because row1 precedes the first divisible row, and the second is divisible. Its balanced symbol sorts to $μ$ , so its coefficient is nonzero. If $i = 1$ , then $σ_{1} = j - 1$ , and the row1 pair at columns $j, j + 1$ has lengths $h, h - 1$ ; its balanced transfer $h - 1$ sorts to $μ$ , again giving a nonzero coefficient. For $η$ , the pair at columns $1, σ_{1} + 1$ has lengths $A^{'} = σ_{1} + h + l - 1$ and $h$ . When $i > 1$ , $A^{'} - h_{i 1}^{μ} = A$ , so $A^{'}$ is nondivisible; when $i = 1$ , $A^{'} = h_{11}^{μ} + h - 1$ , also nondivisible. Its balanced symbol sorts to $η$ . Each claimed target has precisely two lost and two gained beta values, so the uniqueness argument in step 2.2 rules out cancellation. Thus both $J_{ν μ}$ and $J_{ν η}$ are nonzero.

6.1step 1.2step 2.1step 4.1step 2.2step 5.1algebra

Connectivity and sufficiency. Induct upwards from $ρ$ in the finite dominance poset. For $μ \neq ρ$ , if an outgoing coefficient $J_{μ τ}$ is nonzero, then $τ$ is strictly below $μ$ in the same core class by step 2.1, so induction connects $τ$ , and hence $μ$ , to $ρ$ . If no outgoing coefficient exists, steps 2.2, 3.2, 4.2 and 5.1 connect $μ$ to $ν$ and then $η$ , with $η$ strictly below $μ$ , and induction again connects it to $ρ$ . Every such edge is a same-block implication by step 1.2. Thus every pair of labels of the same core lies in the same block. Together with step 4.1 this proves the equivalence.

7.1F1F2F3step 1.1step 4.1step 4.2step 6.1∎

Parameterization and infinite characteristic. Every simple is the head of a Weyl module, so every block has a partition label. The core sizes are $n - e w$ , and each admissible core occurs via the partition $ρ$ of step 4.2; step 6.1 therefore gives the stated indexing. If $e = \infty$ , all coefficients vanish by step 1.1, so positivity forces every first Jantzen submodule, hence every radical, to vanish. All Weyl modules are simple; their decomposition matrix is the identity. Step 4.1 then gives identity Cartan matrix, so every primitive projective is simple and $S$ is semisimple, with each partition its own block and core. This proves all assertions.

Remarks

The connectivity argument follows the corrected dominance-minimum route of Lyle–Mathas Proposition 3.7 and Corollary 4.3, while proving the needed elementary row properties locally. Their final Remark explicitly corrects the earlier first-row-only q-Schur argument; the old direct hook move can have zero coefficient.

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The weight-one block decomposition pattern

Statement

Let $F$ be a field and $q \in F^{\times}$ with finite quantum characteristic $2 \leq e < \infty$ (The quantum characteristic of an invertible Hecke parameter), let $γ$ be an $e$ -core of $m$ , put $n = m + e$ , and fix $d \geq n$ (The e-core, the e-quotient, and the e-weight). There are exactly $e$ partitions $λ_{1}, \dots, λ_{e}$ of $n$ with $e$ -core $γ$ ; list them in dominance order, $λ_{1} ⊳ λ_{2} ⊳ \dots ⊳ λ_{e}$ . In the source's (Murphy) normalisation the Weyl modules and simple modules of this weight-one class satisfy $W^{λ_{e}} = L^{λ_{e}}$ and $[W^{λ_{k}}] = [L^{λ_{k}}] + [L^{λ_{k + 1}}] (1 \leq k < e),$ so the decomposition matrix of the class is the $e \times e$ bidiagonal matrix with rows and columns $λ_{1}, \dots, λ_{e}$ . Moreover exactly $λ_{1}$ is not $e$ -restricted, while $λ_{2}, \dots, λ_{e}$ are. Under the Schur functor (The Schur functor sends Weyl and simple modules to Specht and Hecke simple modules) the corresponding Murphy Specht classes are $[S_{M}^{λ_{1}}] = [D_{M}^{λ_{2}}], [S_{M}^{λ_{k}}] = [D_{M}^{λ_{k}}] + [D_{M}^{λ_{k + 1}}] (2 \leq k < e), [S_{M}^{λ_{e}}] = [D_{M}^{λ_{e}}],$ with $D_{M}^{λ_{1}} = 0$ ; conjugating partitions and applying the same $θ$ pullback to both modules (The type-A Hecke Specht module and its cell form, Ordinary Specht modules are the signed conjugate Murphy cells) gives the corresponding decomposition matrix for the page's Dipper-James Specht modules in $e$ -regular labels.

No choice principle is used.

Facts & Assumptions

Given: A field $F$ , $q \in F^{\times}$ with finite quantum characteristic $2 \leq e < \infty$ , an $e$ -core $γ$ of $m$ , the integer $n = m + e$ , and the Weyl modules $W_{F}^{λ}$ of $S_{F} (d, n)$ for $λ ⊢ n$ with at most $d$ parts, where $d \geq n$ .

[F1]

Each Weyl module has simple head $L^{λ}$ of multiplicity one, the decomposition matrix is dominance-unitriangular, and all simple modules are obtained this way (The type-A q-Schur algebra and its Weyl modules). In a finite-dimensional module, a simple head implies indecomposability: any nonzero direct summand has a nonzero head, so two nonzero summands would give a nonsimple head.

[F2]

The corrected balanced Jantzen formula sums positive filtration submodules, not their successive quotients: $\sum_{i > 0} [W_{F}^{λ} (i)] = \sum_{(a, b), (a, c) \in [λ], b < c} (ν_{π} ([h_{a b}^{λ}]_{\hat{q}}) - ν_{π} ([h_{a c}^{λ}]_{\hat{q}})) [W_{F}^{β + h_{a c}^{λ} ε_{b} - h_{a c}^{λ} ε_{c}}] .$ Here $β_{b} = λ_{b}^{'} + N - b$ for a fixed sufficiently long conjugate beta sequence, and the right hand symbols are alternating symbols: sort strictly decreasing, attach the sorting sign, and vanish on repeated or negative entries. The equal transfer preserves total size. Moreover $W (1) = rad W$ . This is the balanced hook-ratio formula required from The q-Schaper virtual radical-layer formula and Virtual Weyl and Specht symbols attached to beta sequences. Its source arithmetic must be repaired before binding: the manuscript's displayed beta indices are inconsistent, while its worked Example 5.28 determines the balanced transfer.

[F3]

Core-quotient and abacus theory: a weight-one partition is obtained from a packed core abacus by moving the bottom bead of one runner down one place. The bead move removes/adds a rim $e$ -hook, and distinct runners give distinct partitions (The e-core, the e-quotient, and the e-weight, The e-core is independent of the rim hook removal order, Beta-sets, the shift, and the e-runner abacus of a partition). Restricted means $λ_{i} - λ_{i + 1} < e$ for every $i$ (e-regular and e-restricted partitions).

[F4]

The exact Schur functor sends $W^{λ}$ to $S_{M}^{λ}$ and $L^{λ}$ to $D_{M}^{λ}$ , whose nonzero labels are precisely the restricted partitions. The ordinary Specht dictionary is the proved generic- $q$ form bridge and uses conjugation and the same $θ$ pullback on every module (The Schur idempotent and the Schur functor for type-A q-Schur algebras, The Schur functor sends Weyl and simple modules to Specht and Hecke simple modules, The type-A Hecke Specht module and its cell form, Ordinary Specht modules are the signed conjugate Murphy cells).

Proof

technique · direct

1.1F3constructalgebra

Enumeration and ordering. Choose a conjugate core beta set long enough that every runner contains a bead, and list its bottom beads $x_{1} < \dots < x_{e}$ . Take this beta length divisible by $e$ . Every partition with this core has, at the same beta length, the same packed set: its packed set represents $γ$ and the length uniquely determines a beta-set for $γ$ . Weight one means its total runner displacement from packing is one, so exactly one bottom bead moves down one row. Conversely each such move adds exactly $e$ boxes and preserves the core. Thus replacing $x_{k}$ by $x_{k} + e$ gives precisely the $e$ weight-one partitions, denoted $λ_{k}$ . For $j > k$ , moving $x_{j}$ instead of $x_{k}$ gives the more dominant conjugate partition: the two bead lists differ by replacing $x_{k} + e, x_{j}$ with $x_{k}, x_{j} + e$ . The new maximum is larger than both old values and the new minimum smaller than both, with the same sum. For each number of selected entries, the largest possible sum therefore does not decrease, and at least one sorted prefix increases. The sorted conjugate parts differ from these prefixes by the same beta offsets. Conjugation reverses dominance, hence $λ_{1} ⊳ \dots ⊳ λ_{e}$ . For the empty core put $γ_{i} = 0$ . The largest original partition is $(γ_{1} + e, γ_{2}, \dots)$ , since its added horizontal first-row strip dominates every other placement.

1.2F1F2givenalgebra

Choose a lift with valuation one. If $q \neq 1$ , its finite quantum characteristic $e$ is its multiplicative order, and is prime to the field characteristic; use $R = F [X]_{(X - q)}$ and $\hat{q} = X$ . The derivative of $X^{e} - 1$ at $q$ is $e q^{e - 1} \neq 0$ , so $ν ([e]_{X}) = 1$ ; also $ν ([h]_{X}) = 0$ when $e ∤ h$ . If $q = 1$ , then $char F = p = e$ . First work over $F_{p}$ with $R = Z_{(p)}$ , $\hat{q} = 1$ , giving $ν ([e]_{1}) = ν (p) = 1$ and $ν ([h]_{1}) = 0$ for $p ∤ h$ , and then extend scalars to $F$ . This extension preserves the decomposition multiplicities because the q-Schur simple modules are absolutely irreducible and its integral Weyl modules commute with scalar extension. In both cases the generic factorial is nonzero and [F2] applies.

2.1F3step 1.1algebra

Restriction. If $λ$ is not restricted, some $λ_{i} - λ_{i + 1} \geq e$ , so removing the last $e$ boxes of row $i$ leaves a partition. Its size is $∣ γ ∣$ and its core is $γ$ , so it equals $γ$ . Thus $λ$ is obtained by adding $e$ boxes to row $i$ of $γ$ . An $e$ -core has $γ_{r} - γ_{r + 1} < e$ for every $r$ , since otherwise it has a removable horizontal $e$ -strip. Consequently $i > 1$ is impossible: the partition inequality $γ_{i - 1} \geq γ_{i} + e$ would contradict this core property. Therefore the only nonrestricted member is $λ_{1} = (γ_{1} + e, γ_{2}, \dots)$ ; it is indeed nonrestricted because its first-row difference is at least $e$ .

2.2F2F3step 1.1step 1.2algebra

Evaluate the balanced formula. A nonzero coefficient involves a hook of length divisible by $e$ . A weight-one abacus has exactly one bead-gap pair of equal residue with the gap below the bead: $x_{k} + e$ and $x_{k}$ ; thus its only $e$ -divisible hook has length $e$ . Any surviving term in [F2] replaces two bead values, and its new runner counts equal the old ones: either $e$ divides the transfer $h_{a c}$ , leaving residues fixed, or $e$ divides $h_{a b} = h_{a c} + β_{b} - β_{c}$ , exchanging their residues. Hence its sorted symbol is one of the $λ_{j}$ . For $j > k$ , the distinct bead sets of $λ_{k}$ and $λ_{j}$ differ exactly by replacing $x_{k} + e, x_{j}$ with $x_{k}, x_{j} + e$ . There is exactly one balanced transfer yielding this change: if $x_{j} < x_{k} + e$ , its two hook lengths are $h_{a b} = e$ , $h_{a c} = x_{j} - x_{k}$ ; if $x_{j} > x_{k} + e$ , they are $h_{a b} = x_{j} - x_{k}$ , $h_{a c} = e$ . The difference $x_{j} - x_{k}$ is not divisible by $e$ since bottom beads are on distinct runners. Thus the respective unsigned coefficients are $+ 1$ and $- 1$ by step 1.2. Sorting gives in either case the signed coefficient $(- 1)^{j - k + 1}$ : in the packed core set, exactly one bead from each runner with bottom bead $> x_{r}$ lies between $x_{r}$ and $x_{r} + e$ , so moving $x_{r}$ crosses $e - r$ beads; the relative sorting parity for the two moves, together with their opposite orientation, is $(- 1)^{(e - j) - (e - k) + 1}$ . There are no other terms and no repeated contributions: the two lost and two gained bead values determine the pair and transfer uniquely. Therefore $\sum_{i > 0} [W^{λ_{k}} (i)] = \sum_{j > k} (- 1)^{j - k + 1} [W^{λ_{j}}]$ .

3.1F1F2step 2.2algebra

Descending induction and the matrix. For $k = e$ , step 2.2 gives zero on the left; all summands are actual submodule classes with nonnegative simple multiplicities, so $W^{λ_{e}} (1) = 0$ and $W^{λ_{e}} = L^{λ_{e}}$ . Suppose the stated rows are known for $j > k$ . Substitution into step 2.2 telescopes to $\sum_{i > 0} [W^{λ_{k}} (i)] = [L^{λ_{k + 1}}]$ . Since $W (1) = rad W$ and each later term is a submodule of $W (1)$ , positivity shows $W (1) ≅ L^{λ_{k + 1}}$ and $W (i) = 0$ for $i \geq 2$ . The head quotient is $L^{λ_{k}}$ , giving $[W^{λ_{k}}] = [L^{λ_{k}}] + [L^{λ_{k + 1}}]$ . Notice that the identity $[W] = [L] + \sum_{i > 0} [W (i)]$ is used only after proving all later filtration submodules vanish; it is false for a general Jantzen filtration. Reading simple multiplicities yields the complete $e \times e$ bidiagonal matrix.

4.1F4step 2.1step 3.1∎

Specht rows. Apply the exact functor of [F4] to step 3.1 and use step 2.1: $D_{M}^{λ_{1}} = 0$ , while the other $D_{M}^{λ_{k}}$ are the nonzero simples. This gives exactly the three displayed Murphy Specht class identities, hence the full $e \times (e - 1)$ Murphy matrix. Conjugation sends the core to $γ^{'}$ and the restricted simple labels to regular ones, and the same $θ$ pullback applied to every module preserves all composition multiplicities. Thus the ordinary Specht matrix has the same entries under the accepted conjugate/sign label dictionary.

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Type-A Hecke Specht modules are in the same block iff their labels have the same e-core

Statement

Let $F$ be a field and $q \in F^{\times}$ with quantum characteristic $e$ (The quantum characteristic of an invertible Hecke parameter), and let $S^{λ}$ be the type-A Hecke Specht module of this page in the Dipper-James convention (The type-A Hecke Specht module and its cell form) for $λ ⊢ n$ . Then for all partitions $λ, μ ⊢ n$ the modules $S^{λ}$ and $S^{μ}$ lie in the same block of $H_{F, q} (S_{n})$ if and only if $λ$ and $μ$ have the same $e$ -core (The e-core, the e-quotient, and the e-weight). Equivalently, for finite $e$ , the blocks of $H_{F, q} (S_{n})$ are indexed by the $e$ -cores $γ$ of size at most $n$ with $∣ γ ∣ \equiv n (m o d e)$ , and the block of $S^{λ}$ is determined by $C_{e} (λ)$ . For $e = \infty$ use $C_{\infty} (λ) : = λ$ ; the Hecke algebra is semisimple and its blocks are indexed by partitions of $n$ .

No choice principle is used.

Facts & Assumptions

Given: A field $F$ , $q \in F^{\times}$ with quantum characteristic $e$ , an integer $n \geq 0$ , the Hecke algebra $H = H_{F, q} (S_{n})$ with Specht modules $S^{λ}$ for $λ ⊢ n$ , and a fixed integer $d \geq n$ .

[F1]

For $d \geq n$ , the Schur idempotent $e \in S = S_{F} (d, n)$ satisfies $e S e ≅ H$ , and $X \mapsto X e$ is exact. It sends each Weyl module $W^{λ}$ to the nonzero Murphy cell module $S_{M}^{λ}$ ; its simple images are zero or the Murphy simple modules, with the nonzero images indexed by the $e$ -restricted partitions (The Schur idempotent and the Schur functor for type-A q-Schur algebras, The Schur functor sends Weyl and simple modules to Specht and Hecke simple modules).

[F2]

Blocks: $1 = \sum_{D} δ_{D}$ is the unique decomposition of the identity of a finite-dimensional algebra into pairwise orthogonal primitive central idempotents, blocks are the ideals cut out by the $δ_{D}$ , and every nonzero central idempotent is a sum of some of the $δ_{D}$ (Blocks of a finite-dimensional algebra via primitive central idempotents, The identity of a finite-dimensional algebra decomposes into primitive central idempotents, Decompositions of the identity into primitive central idempotents are unique).

[F3]

The q-Schur algebra and its Weyl modules: $S$ is finite-dimensional, $S = {E n d}_{H} (M)$ for $M = ⨁_{ν \in Λ (d, n)} M^{ν}$ with $M^{(1^{n})} = H$ , and $W_{F}^{λ}$ has head $L_{F}^{λ}$ (The type-A q-Schur algebra and its Weyl modules).

[F4]

The q-Schur block theorem q-Schur Weyl modules are in the same block iff their labels have the same e-core: for $λ, μ ⊢ n$ , the Weyl modules $W_{F}^{λ}, W_{F}^{μ}$ lie in the same block of $S$ if and only if $C_{e} (λ) = C_{e} (μ)$ ; the blocks of $S$ are indexed by the $e$ -cores $γ$ with $∣ γ ∣ \leq n$ , $∣ γ ∣ \equiv n (m o d e)$ ; and if $e = \infty$ then $S$ is semisimple with each $W_{F}^{λ}$ its own block.

[F5]

The accepted ordinary/Murphy dictionary identifies the ordinary Specht module with the fixed $θ$ pullback of the conjugate-shape Murphy cell module. Conjugation sends $C_{e} (λ)$ to $C_{e} (λ)^{'}$ , and sends restricted labels to regular labels. In particular it preserves equality of cores. An algebra automorphism permutes primitive central idempotents and therefore preserves the same-block relation (The type-A Hecke Specht module and its cell form).

Proof

technique · direct

1.1F1F2F3

Central idempotents of $H$ from blocks of $S$ . For a block $C$ of $S$ with primitive central idempotent $ϵ_{C}$ put ${\tilde{ϵ}}_{C} : = ϵ_{C} e = e ϵ_{C} e \in H$ . This element is a central idempotent of $H$ : it is idempotent because $ϵ_{C}$ is, and for $x = e s e \in H$ centrality of $ϵ_{C}$ gives $x {\tilde{ϵ}}_{C} = e s ϵ_{C} e = e ϵ_{C} s e = {\tilde{ϵ}}_{C} x$ ; it is nonzero because it acts as the identity on the nonzero module $W_{F}^{λ} e = S_{M}^{λ}$ for any $λ$ with $W_{F}^{λ} \in C$ , namely $S_{M}^{λ} {\tilde{ϵ}}_{C} = W_{F}^{λ} e ϵ_{C} e = W_{F}^{λ} ϵ_{C} e = S_{M}^{λ}$ . Moreover ${\tilde{ϵ}}_{C} {\tilde{ϵ}}_{C^{'}} = e ϵ_{C} ϵ_{C^{'}} e = 0$ for $C \neq C^{'}$ and $\sum_{C} {\tilde{ϵ}}_{C} = e \sum_{C} ϵ_{C} e = e^{2} = 1_{H}$ .

1.2F2F3algebra

Number of q-Schur blocks is at least the number of Hecke blocks. Let $H = ⨁_{i} B_{i}$ be the block decomposition of $H$ with block identities $b_{i}$ . Since $M = ⨁_{ν} M^{ν}$ and $H = M^{(1^{n})}$ is a direct summand, $M = ⨁_{i} M b_{i}$ with each $M b_{i} \neq 0$ (it contains $H b_{i} = B_{i}$ ), and ${H o m}_{H} (M b_{i}, M b_{j}) = 0$ for $i \neq j$ , because a map $f$ satisfies $f (x) = f (x b_{i}) = f (x) b_{i} \in M b_{j} b_{i} = 0$ . Hence $S = {E n d}_{H} (M) = ⨁_{i, j} {H o m}_{H} (M b_{i}, M b_{j}) = \prod_{i} {E n d}_{H} (M b_{i})$ is a product of the nonzero algebras ${E n d}_{H} (M b_{i})$ , and a product of nonzero algebras has at least one block per factor; so $# {blocks of S} \geq # {blocks of H}$ .

2.1F2step 1.1

Number of Hecke blocks is at least the number of q-Schur blocks. Decompose each nonzero central idempotent ${\tilde{ϵ}}_{C}$ of step 1.1 into primitive central idempotents of $H$ using the unique decomposition of $1_{H}$ : every primitive central idempotent $δ_{D}$ of $H$ with $δ_{D} {\tilde{ϵ}}_{C} = δ_{D}$ belongs to exactly one ${\tilde{ϵ}}_{C}$ , because $δ_{D} = δ_{D} \cdot 1_{H} = \sum_{C} δ_{D} {\tilde{ϵ}}_{C}$ and pairwise orthogonality of the $δ_{D}$ forces $δ_{D} {\tilde{ϵ}}_{C} \in {0, δ_{D}}$ . Since each ${\tilde{ϵ}}_{C} \neq 0$ has at least one such $δ_{D}$ , the blocks $D$ of $H$ are partitioned into nonempty supports over the blocks $C$ of $S$ , and therefore $# {blocks of H} \geq # {blocks of S}$ .

3.1F1F4step 2.1step 1.2

A bijection of block sets, and the core criterion. By steps 2.1 and 1.2 the two block counts are equal, so each ${\tilde{ϵ}}_{C}$ of step 1.1 has exactly one primitive summand and the map $C \mapsto {\tilde{ϵ}}_{C}$ is a bijection from the blocks of $S$ onto the blocks of $H$ . Since $S_{M}^{λ} = W_{F}^{λ} e$ and $S_{M}^{μ} = W_{F}^{μ} e$ , the modules $S_{M}^{λ}$ and $S_{M}^{μ}$ lie in the same Hecke block if and only if $W_{F}^{λ}$ and $W_{F}^{μ}$ lie in the same q-Schur block: indeed the block of $S_{M}^{λ}$ is the image under the bijection of the block of $W_{F}^{λ}$ . By the block theorem [F4] the latter holds if and only if $C_{e} (λ) = C_{e} (μ)$ , which proves the q-Schur-normalised statement.

4.1F1F4F5step 3.1algebra∎

Ordinary normalisation and semisimplicity. By [F5], $S^{λ}$ and $S^{μ}$ lie in the same block exactly when $S_{M}^{λ^{'}}$ and $S_{M}^{μ^{'}}$ do. Step 3.1 makes this equivalent to $C_{e} (λ^{'}) = C_{e} (μ^{'})$ , and the identity $C_{e} (ν^{'}) = C_{e} (ν)^{'}$ makes it equivalent to $C_{e} (λ) = C_{e} (μ)$ . This identity follows by conjugating every removable border strip in a core-removal sequence; the terminal core also conjugates, by independence of removal order. For $e = \infty$ , $S$ is semisimple by [F4]. Its corner $e S e$ is semisimple: by Wedderburn–Artin theorem for semisimple rings, in the Wedderburn decomposition of $S$ , each idempotent selects a subspace in each matrix factor, and its corner is a product of matrix algebras over the same division rings. Thus $H ≅ e S e$ is semisimple; the block bijection separates the partition labels. This proves all assertions.

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Nakayama block classification for symmetric groups

Statement

Let $k$ be a splitting field of characteristic $p > 0$ (equivalently the residue field of the fixed splitting $p$ -modular system, A splitting p-modular system for a finite group is a p-modular system whose fraction and residue fields split the needed group algebras, p-blocks from primitive central idempotents) and let $n \geq 0$ . Two Specht modules $S_{k}^{λ}, S_{k}^{μ}$ of $k [S_{n}]$ lie in the same block of $k [S_{n}]$ if and only if $λ, μ ⊢ n$ have the same $p$ -core (The e-core, the e-quotient, and the e-weight). Consequently the blocks of $k [S_{n}]$ are indexed by the pairs $(γ, w)$ with $γ$ a $p$ -core, $w \in N_{0}$ and $n = ∣ γ ∣ + p w$ , and the block of $S_{k}^{λ}$ is the pair $(C_{p} (λ), w_{p} (λ))$ with $w_{p} (λ) = (n - ∣ C_{p} (λ) ∣) / p$ (The core-quotient size formula defines the weight).

No choice principle is used.

Facts & Assumptions

Given: A splitting field $k$ of characteristic $p > 0$ with the fixed splitting $p$ -modular system, an integer $n \geq 0$ , and the group algebra $k [S_{n}]$ with group Specht modules $S_{k}^{λ}$ for $λ ⊢ n$ .

[F1]

The algebra isomorphism $H_{k, 1} (S_{n}) ≅ k [S_{n}]$ sends each ordinary Dipper-James Specht module to the classical group Specht module of the same shape, and the quantum characteristic at $q = 1$ in characteristic $p$ equals $p$ (Hecke Specht theory specialises to group Specht theory at q = 1, The quantum characteristic of an invertible Hecke parameter). The module comparison uses the accepted signed conjugate Murphy/ordinary dictionary before specialising; it does not identify the same-shape Murphy cell module with the ordinary Specht module.

[F2]

The Hecke block criterion, in the Dipper-James normalisation of the same Specht modules as [F1]: Type-A Hecke Specht modules are in the same block iff their labels have the same e-core states that for a field $k$ and $q \in k^{\times}$ of quantum characteristic $e \geq 2$ , two Specht modules $S^{λ}, S^{μ}$ of $H_{k, q} (S_{n})$ lie in the same block if and only if $C_{e} (λ) = C_{e} (μ)$ , and that the blocks are indexed by the $e$ -cores $γ$ with $∣ γ ∣ \leq n$ and $∣ γ ∣ \equiv n (m o d e)$ .

[F4]

The core-quotient size formula The core-quotient size formula defines the weight: for every $e \geq 2$ and $λ ⊢ n$ one has $n = ∣ C_{e} (λ) ∣ + e w_{e} (λ)$ with $w_{e} (λ) = ∣ Q_{e} (λ) ∣ \in N_{0}$ , and for an $e$ -core $γ$ the condition $∣ γ ∣ \leq n$ , $∣ γ ∣ \equiv n (m o d e)$ is exactly the condition that $n - ∣ γ ∣$ be an admissible $e$ -weight; conversely every such $γ$ occurs as the core of a partition of $n$ (q-Schur Weyl modules are in the same block iff their labels have the same e-core, The e-core is independent of the rim hook removal order).

[F5]

An algebra isomorphism $φ : A \to B$ preserves blocks: it maps a central idempotent $z$ to the central idempotent $φ (z)$ , preserves primitivity, and hence induces a bijection between the block sets (Blocks of a finite-dimensional algebra via primitive central idempotents, Decompositions of the identity into primitive central idempotents are unique).

Proof

technique · direct

1.1F1F5algebra

Specialisation. Take $q = 1 \in k^{\times}$ and $e$ the quantum characteristic of $(k, q)$ ; since $q = 1$ and $char k = p > 0$ , [F1] gives $e = p$ . By [F1] there is an algebra isomorphism $H_{k, 1} (S_{n}) ≅ k [S_{n}]$ carrying the Hecke Specht modules of $H_{k, 1} (S_{n})$ to the group Specht modules $S_{k}^{λ}$ , so by [F5] this isomorphism induces a bijection between the blocks of the two algebras that matches $S^{λ}$ with $S_{k}^{λ}$ ; in particular $S_{k}^{λ}$ and $S_{k}^{μ}$ lie in the same block of $k [S_{n}]$ if and only if the corresponding Hecke Specht modules lie in the same block of $H_{k, 1} (S_{n})$ .

2.1F2step 1.1

The core criterion. Applying the Hecke block criterion [F2] with $q = 1$ and $e = p$ , the Hecke Specht modules of $H_{k, 1} (S_{n})$ lie in the same block if and only if the labels $λ, μ$ have the same $p$ -core. Together with step 1.1 this gives: $S_{k}^{λ}$ and $S_{k}^{μ}$ lie in the same block of $k [S_{n}]$ if and only if $C_{p} (λ) = C_{p} (μ)$ .

3.1F4step 2.1algebra

The parametrisation by cores and weights. By [F2] the blocks of $H_{k, 1} (S_{n})$ , hence by step 1.1 the blocks of $k [S_{n}]$ , are indexed by the $p$ -cores $γ$ with $∣ γ ∣ \leq n$ and $∣ γ ∣ \equiv n (m o d p)$ ; the block of $S_{k}^{λ}$ corresponds to $C_{p} (λ)$ . For $n \geq 0$ and such a $γ$ put $w : = (n - ∣ γ ∣) / p \in N_{0}$ ; by [F4] this is exactly the $p$ -weight $w_{p} (λ)$ of any Specht label with core $γ$ , and $n = ∣ γ ∣ + p w$ . Since $w$ is determined by $n$ and $γ$ , the map $γ \mapsto (γ, w)$ is a bijection onto the displayed set of pairs, and the block of $S_{k}^{λ}$ is the pair $(C_{p} (λ), w_{p} (λ))$ .

4.1step 2.1step 3.1∎

Steps 2.1 and 3.1 establish both assertions: the same-block criterion by the $p$ -core, and the indexing of the blocks by the pairs $(γ, w)$ with $n = ∣ γ ∣ + p w$ together with the identification of the block of $S_{k}^{λ}$ as $(C_{p} (λ), w_{p} (λ))$ .

The classification is stated independently, by a Young-module and Brauer-quotient route, as Theorem 1 of O'Donovan's preprint recorded in the sources; that reference is an independent confirmation of the statement and is not used in the proof above. No choice principle is used anywhere in this item.

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

Weight-zero symmetric-group blocks are defect zero

Statement

Let $k$ be a splitting field of characteristic $p > 0$ and let $λ ⊢ n$ be a partition of $p$ -weight $w_{p} (λ) = 0$ , that is, a $p$ -core (The e-core, the e-quotient, and the e-weight). Then the block $B$ of $k [S_{n}]$ containing $S_{k}^{λ}$ (Nakayama block classification for symmetric groups) contains exactly one Specht label, namely $λ$ ; its decomposition matrix is the $1 \times 1$ matrix $(1)$ , so $D^{λ}$ is both the unique simple and the unique projective indecomposable module of $B$ ; its Cartan matrix is $(1)$ ; the block algebra is a full matrix algebra over $k$ ; and $B$ has trivial defect group, that is, defect zero in the fixed splitting modular system (Defect group and numerical defect of a block).

No choice principle is used.

Facts & Assumptions

Given: A splitting field $k$ of characteristic $p > 0$ for $S_{n}$ and all its subgroups, an integer $n \geq 0$ , a partition $λ ⊢ n$ with $w_{p} (λ) = 0$ , and the block $B$ of $k [S_{n}]$ containing $S_{k}^{λ}$ .

[F1]

$w_{p} (λ) = (n - ∣ C_{p} (λ) ∣) / p$ , so $w_{p} (λ) = 0$ is equivalent to $C_{p} (λ) = λ$ , and a $p$ -core is $p$ -regular (The e-core, the e-quotient, and the e-weight, The core-quotient size formula defines the weight). Hence $D^{λ} \neq 0$ and $S_{k}^{λ}$ has simple head $D^{λ}$ , with $D^{λ}$ its unique maximal quotient, so $S_{k}^{λ}$ is indecomposable (Modular simple modules of the symmetric group).

[F2]

Nakayama: two Specht modules of $k [S_{n}]$ lie in the same block if and only if their labels have the same $p$ -core (Nakayama block classification for symmetric groups); the blocks are indexed by the $p$ -cores $γ$ with $∣ γ ∣ \leq n$ , $∣ γ ∣ \equiv n (m o d p)$ . The simple $k [S_{n}]$ -modules are exactly the $D^{μ}$ for the $p$ -regular $μ ⊢ n$ , pairwise non-isomorphic (Modular simple modules of the symmetric group).

[F3]

Block bookkeeping: the identity of $k [S_{n}]$ decomposes uniquely into primitive central idempotents, every finite-dimensional module decomposes along the blocks, and a simple module lies in exactly one block (p-blocks from primitive central idempotents, Blocks of a finite-dimensional algebra via primitive central idempotents); in particular an indecomposable finite-dimensional module lies in a single block.

[F4]

Dominance unitriangularity: $d_{μ μ} = 1$ and $d_{μ ν} \neq 0$ implies $ν ⊵ μ$ for the modular decomposition numbers $d_{μ ν} = [S_{k}^{μ} : D^{ν}]$ (Dominance unitriangularity of the symmetric-group decomposition matrix).

[F5]

For a block $B$ with decomposition matrix $D_{B}$ and Cartan matrix $C_{B}$ one has $C_{B} = D_{B}^{T} D_{B}$ (The Cartan matrix is D^T D); the $(1, 1)$ -entry of $C_{B}$ is the multiplicity of the simple module in its projective cover (Projective indecomposable characters and Cartan invariants).

[F6]

Defect-zero criterion: over the splitting residue field, for a block $B$ the following are equivalent: the defect group of $B$ is trivial; $B$ is a full matrix algebra over $k$ ; $B$ has a projective simple module; in that case there is exactly one simple module and it is projective (Defect zero blocks are simple algebras).

Proof

technique · direct

1.1F1F2F3F4

Bookkeeping for the labels of $B$ . Let $ν ⊢ n$ be a $p$ -regular partition with block $B (ν) = B$ ; by [F1] and [F2] this is equivalent to $C_{p} (ν) = C_{p} (λ) = λ$ , and since $∣ ν ∣ = n = ∣ λ ∣$ this forces $ν = λ$ . Hence the only $p$ -regular label in $B$ is $λ$ , and by [F2] the only simple module in $B$ is $D^{λ}$ . For any Specht label $μ$ in $B$ , [F2] gives $C_{p} (μ) = λ$ and $∣ μ ∣ = ∣ λ ∣$ , so $μ = λ$ as well: the block has exactly one Specht row. Since $S_{k}^{λ}$ is indecomposable with head $D^{λ}$ by [F1] and lies in $B$ by hypothesis, the decomposition matrix of $B$ is the $1 \times 1$ matrix with entry $d_{λ λ} = [S_{k}^{λ} : D^{λ}] = 1$ , by the exact diagonal multiplicity in [F4]. A simple head alone would not establish multiplicity one.

2.1F1F5step 1.1algebra

The Cartan matrix and the projective cover. By [F5] applied to the block $B$ the Cartan matrix of $B$ is $C_{B} = D_{B}^{T} D_{B} = (1)$ , so the projective cover $P^{λ}$ of $D^{λ}$ has $D^{λ}$ as its only composition factor, with multiplicity one. A nonzero module of length one is simple, so $P^{λ} ≅ D^{λ}$ : the block $B$ has a projective simple module.

3.1F5F6step 1.1step 2.1

Defect zero and the matrix algebra. By step 2.1 the projective indecomposable $P^{λ}$ equals the simple $D^{λ}$ , and it is the unique projective indecomposable of $B$ because the simple modules of $B$ are in bijection with its projective indecomposables; so $D^{λ}$ is both the unique simple and the unique projective indecomposable of $B$ . The defect-zero criterion [F6] now gives that $B$ is a full matrix algebra over $k$ and that the defect group of $B$ is trivial, that is, $B$ has defect zero in the fixed splitting modular system.

4.1step 1.1step 2.1step 3.1∎

Steps 1.1, 2.1 and 3.1 give all the assertions: the single Specht label $λ$ , the decomposition matrix $(1)$ , the unique simple and projective indecomposable $D^{λ}$ , the Cartan matrix $(1)$ , the full matrix algebra structure and defect zero.

RemarkRemark: AI-adaptedProof: Not applicableOpen item page →

Block weight is not the same as numerical defect

Remark

Weight is combinatorial. For finite $e \geq 2$ , the $e$ -weight $w_{e} (λ)$ of a label $λ ⊢ n$ is the number of $e$ -hooks that have to be removed from $λ$ to reach its $e$ -core, it equals the size $∣ Q_{e} (λ) ∣$ of the $e$ -quotient, and it satisfies $n = ∣ λ ∣ = ∣ C_{e} (λ) ∣ + e w_{e} (λ)$ (The core-quotient size formula defines the weight). In particular, $w_{e} (λ) = (n - ∣ C_{e} (λ) ∣) / e$ is determined by $n$ and the $e$ -core. Thus, for a fixed algebra $H_{F, q} (S_{n})$ , by Type-A Hecke Specht modules are in the same block iff their labels have the same e-core it is an invariant of the block of $S^{λ}$ ; for $e = p$ it is the combinatorial $p$ -weight of the block.

Numerical defect is block-theoretic. The numerical defect of a block is defined from its defect group: for the block bimodule $k G b$ a defect group is a $p$ -subgroup $D \leq G$ for which the diagonal $Δ D$ is a vertex, its order $p^{d}$ defines the numerical defect $d$ , and defect zero means $D = 1$ (Defect group and numerical defect of a block). It is a $p$ -adic order invariant of the block algebra in the fixed splitting modular system and is defined by a vertex computation, not by hook combinatorics.

The two invariants must not be identified without a theorem and an explicit range. The weight records how many hooks of length divisible by $e$ the label carries, while the numerical defect records the order of a vertex of the block bimodule; nothing on this page proves that the two numbers agree, and they are defined from entirely different data. The only case settled here is weight zero: if $w_{p} (λ) = 0$ then $λ$ is a $p$ -core, and the classification forces the block to have a single Specht label, Cartan matrix $(1)$ , a projective simple module, a full matrix algebra structure and trivial defect group, so the numerical defect is also zero (Weight-zero symmetric-group blocks are defect zero). For $w > 0$ no assertion about the defect group of a weight- $w$ symmetric-group block, and no identification of $w$ with the numerical defect, is made on this page. The printed defect-group computation for weight- $w$ blocks (defect $ν_{p} ((p w)!)$ , with defect group a Sylow $p$ -subgroup of $S_{p w}$ ) is deliberately not asserted here, because no counted proof source for it is registered with this batch; supplying it is outside this page's proof route and remains an open obligation for a later page of the track.

No choice principle is used in the statements recorded above.

5 · Examples, counterexamples and false statements

None yet.

Cores, Quotients, and Blocks of Symmetric Groups

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Beta-sets, the shift, and the e-runner abacus of a partition

Definition

Remarks

Hooks are bead-gap pairs on the abacus

Statement

Facts & Assumptions

Proof

The e-core, the e-quotient, and the e-weight

Definition

Remarks

Virtual Weyl combinations detect q-Schur block separation

Statement

Facts & Assumptions

Proof

The e-core is independent of the rim hook removal order

Statement

Facts & Assumptions

Proof

The e-quotient controls the hooks of e-divisible length

Statement

Facts & Assumptions

Proof

The core-quotient size formula defines the weight

Statement

Facts & Assumptions

Proof

q-Schur Weyl modules are in the same block iff their labels have the same e-core

Statement

Facts & Assumptions

Proof

Remarks

The weight-one block decomposition pattern

Statement

Facts & Assumptions

Proof

Type-A Hecke Specht modules are in the same block iff their labels have the same e-core

Statement

Facts & Assumptions

Proof

Nakayama block classification for symmetric groups

Statement

Facts & Assumptions

Proof

Weight-zero symmetric-group blocks are defect zero

Statement

Facts & Assumptions

Proof

Block weight is not the same as numerical defect

Remark

5 · Examples, counterexamples and false statements

Sources