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KLR Algebras, Induction, and Quantum-Serre Categorification

1 · Prerequisites

2 · Summary

Objective. This page builds the generic theory of quiver Hecke (KLR) algebras and their categorification of the negative half of a symmetrizable quantum group. It starts from the polynomial presentation of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum and the internal grading of The KLR grading and the diagrammatic multiplication, proves the faithful divided-difference polynomial representation Faithful polynomial representation of a KLR algebra, and deduces the PBW (monomial) basis theorem The PBW (monomial) basis theorem for KLR algebras with finite-dimensional graded components. On that foundation it defines the induction and restriction bimodules Induction and restriction bimodules for KLR algebras, proves exactness, projective preservation and the two adjunctions (left $I n d ⊣ R e s$ , right coinduction as reversed-block induction) together with the Mackey filtration in Generic KLR induction and restriction adjunctions preserve projectives, constructs the nil-Hecke divided-power idempotents and projectives Nil-Hecke idempotents and divided-power KLR projectives, and records the local graded Krull-Schmidt and shift-orbit basis facts Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules and the maximal-tail/unique-head/absolute-simplicity package Maximal KLR tails give a unique induction head and absolute simple modules.

Quantum Serre and the Cartan form. The induction product on the split Grothendieck group satisfies the divided-power quantum Serre relations (The KLR Grothendieck ring satisfies the divided-power quantum Serre relations), so it receives the isomorphism $γ$ from the divided-power form of the negative half. The graded Hom (Cartan) pairing is defined in The KLR Cartan (graded Hom) pairing on the projective Grothendieck group and identified with the Lusztig-normalized half form in The KLR Cartan pairing is the Lusztig-normalized half form; the combined statement is the categorification theorem The KLR category categorifies the negative quantum half, an isomorphism of graded twisted bialgebras with compatible coinvolutions.

Scope and guards. All of the above holds for every field and every graded hermitian KLR parameter satisfying (Q1)-(Q4). The final global-basis identification Self-dual indecomposable KLR projectives give the lower global basis (symmetric type) is stated only for symmetric Cartan data with $d_{i} = 1$ , characteristic zero and finite loopless quiver parameters $Q_{i j} = (- 1)^{h_{i j}} (u - v)^{h_{i j} + h_{j i}}$ , with arbitrary arrow multiplicities; it consumes the geometric suppliers on signed-quiver-flag-convolution-and-klr-ext-realization and quiver-full-flag-projective-semisimplicity and the positive first-string supplier Self-dual KLR projectives have positive first-block divided-string leading terms. The companion page klr-algebras-induction-and-quantum-serre-categorification-examples works the small cases, the double-edge braid correction, the degenerate-parameter boundary and the rank-two nil-Hecke projective.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum

Definition

Let $(I, A, D, P, P^{\lor}, q)$ be a symmetrizable Cartan datum with no imaginary simple roots, so $a_{i i} = 2$ , and put $b_{i j} : = d_{i} a_{i j} = (α_{i} ∣ α_{j})$ , a symmetric matrix with $b_{i i} = 2 d_{i} > 0$ (Symmetrizable Cartan data for quantum groups, Kac Moody root lattice height and positive cone). Fix a field $k$ . A KLR datum on it is a family $Q = (Q_{i j})_{i \neq j}$ of polynomials $Q_{i j} \in k [u, v]$ such that

(Q1) $Q_{i j} \neq 0$ for $i \neq j$ and $Q_{i i} : = 0$ ;

(Q2) $Q_{i j} (u, v) = Q_{j i} (v, u)$ for all $i \neq j$ ;

(Q3) $Q_{i j}$ is homogeneous of total degree $- 2 b_{i j}$ for the degree assignments $\deg u : = 2 d_{i}$ , $\deg v : = 2 d_{j}$ ;

(Q4) the two extremal coefficients of $Q_{i j}$ , namely that of $u^{- a_{i j}}$ and that of $v^{- a_{j i}}$ , are nonzero.

For a chosen finite quiver with a compatible automorphism having no arrow within a vertex orbit, write $i \cdot j = b_{i j}$ and let $d_{i j}$ be the number of arrow orbits directed from vertex orbit $i$ to vertex orbit $j$ ; the specialization applies when its induced Cartan datum is the fixed one. Then one takes $Q_{i j} (u, v) = (- 1)^{d_{i j}} (u^{l / (i \cdot i)} - v^{l / (j \cdot j)})^{- 2 (i \cdot j) / l}$ with $l = l c m (i \cdot i, j \cdot j)$ , and $Q_{i j} = 1$ when $a_{i j} = 0$ ; the cyclic affine $A_{1}^{(1)}$ double edge $a_{i j} = a_{j i} = - 2$ is the instance $Q_{i j} (u, v) = - (u - v)^{2}$ up to the fixed normalization of the top coefficients. The simply-laced standard choice is $Q_{i j} (u, v) = u + v$ for $a_{i j} = - 1$ .

The algebra. For $n \geq 0$ let $I^{n}$ be the set of sequences, let $e (ν)$ for $ν \in I^{n}$ be idempotents, and let $x_{a}$ ( $1 \leq a \leq n$ ) and $τ_{a}$ ( $1 \leq a \leq n - 1$ ) be generators. The KLR algebra $R_{n} (Q)$ is the unital associative $k$ -algebra generated by the $e (ν), x_{a}, τ_{a}$ subject to the relations, for all $ν \in I^{n}$ (with $s_{a}$ the adjacent transposition acting on positions $a, a + 1$ ):

(1) $e (ν) e (ν^{'}) = δ_{ν ν^{'}} e (ν)$ and $\sum_{ν \in I^{n}} e (ν) = 1$ ;

(2) $x_{a} e (ν) = e (ν) x_{a}$ and $x_{a} x_{b} = x_{b} x_{a}$ ;

(3) $τ_{a} e (ν) = e (s_{a} ν) τ_{a}$ ;

(4) $τ_{a}^{2} e (ν) = Q_{ν_{a} ν_{a + 1}} (x_{a}, x_{a + 1}) e (ν)$ ;

(5) $τ_{a} τ_{b} = τ_{b} τ_{a}$ whenever $∣ a - b ∣ > 1$ ;

(6) $(τ_{a} x_{b} - x_{s_{a} (b)} τ_{a}) e (ν) = 0$ if $b \notin {a, a + 1}$ or $ν_{a} \neq ν_{a + 1}$ , while if $ν_{a} = ν_{a + 1}$ one has $(τ_{a} x_{a} - x_{a + 1} τ_{a}) e (ν) = e (ν)$ and $(τ_{a} x_{a + 1} - x_{a} τ_{a}) e (ν) = - e (ν)$ (the sign normalization of the divided difference $τ_{a} (f) = (f - s_{a} f) / (x_{a} - x_{a + 1})$ ; Rouquier's printed relation has the opposite global sign, the two conventions being exchanged by $τ_{a} \mapsto - τ_{a}$ );

(7) $(τ_{a + 1} τ_{a} τ_{a + 1} - τ_{a} τ_{a + 1} τ_{a}) e (ν) = 0$ if $ν_{a} \neq ν_{a + 2}$ , and $(τ_{a + 1} τ_{a} τ_{a + 1} - τ_{a} τ_{a + 1} τ_{a}) e (ν) = \frac{Q_{ν_{a} ν_{a + 1}} (x_{a}, x_{a + 1}) - Q_{ν_{a} ν_{a + 1}} (x_{a + 2}, x_{a + 1})}{x_{a + 2} - x_{a}} e (ν)$ if $ν_{a} = ν_{a + 2}$ , the numerator carrying the sign of the $τ$ -normalization of (6) (Rouquier's printed numerator is its negative); the quotient is a polynomial in $x_{a}, x_{a + 1}, x_{a + 2}$ because $Q_{i j}$ is a polynomial in its first variable and the numerator vanishes at $x_{a + 2} = x_{a}$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, Polynomial rings in finitely many commuting indeterminates by iteration).

For $β \in Q_{+}$ put $I^{β} : = {ν \in I^{∣ β ∣} : \sum_{a} α_{ν_{a}} = β}$ , $e (β) : = \sum_{ν \in I^{β}} e (ν)$ and $R (β) : = e (β) R_{∣ β ∣} (Q) e (β) = ⨁_{ν, η \in I^{β}} e (ν) R_{∣ β ∣} (Q) e (η)$ , and put $R : = ⨁_{β \in Q_{+}} R (β)$ . The presentation is the sequence form of Rouquier's $H_{n} (Q)$ via $1_{ν} = e (ν)$ , $x_{a, ν} = x_{a} e (ν)$ , $τ_{a, ν} = τ_{a} e (ν)$ ; no sign or orientation beyond $Q$ is part of the datum, and $R (0) = k$ .

Remarks

Well-definedness. $R_{n} (Q)$ is defined as the quotient of the free unital associative $k$ -algebra on the symbols $e (ν)$ ( $ν \in I^{n}$ ), $x_{a}$ , $τ_{a}$ by the two-sided ideal generated by the displayed relations, so no existence issue arises. Two points deserve record. First, relation (7) is stated as an equality in $k [x_{a}, x_{a + 1}, x_{a + 2}] e (ν)$ : writing $Q (u, v) = \sum_{a} u^{a} q_{a} (v)$ , the difference $Q (u, v) - Q (w, v) = \sum_{a} (u^{a} - w^{a}) q_{a} (v)$ is divisible by $u - w$ in the polynomial ring, and its quotient evaluated at $(u, w) = (x_{a}, x_{a + 2})$ , $v = x_{a + 1}$ gives the displayed correction; the special case $a_{i j} = 0$ with $Q_{i j}$ a nonzero constant is included, where the correction vanishes and the ordinary braid relation holds on $ν_{a} = ν_{a + 2}$ . Second, the mixed relations are word-component identities: the final $e (ν)$ is essential, since a global dot commutator has different components on equal and unequal colors. On an equal-color component they are the identities $\partial_{a} x_{a} - x_{a + 1} \partial_{a} = 1$ and $\partial_{a} x_{a + 1} - x_{a} \partial_{a} = - 1$ for $\partial_{a} = (1 - s_{a}) / (x_{a} - x_{a + 1})$ . The two global sign conventions for $τ_{a}$ are exchanged by $τ_{a} \mapsto - τ_{a}$ , which preserves (4) and flips the signs in (6) and (7); this page fixes the convention of (6) once and for all and every later item uses it.

Choice. The definitional clauses are choice-free: they name a presented algebra and its weight decomposition over a fixed datum, with no selection of representatives or of auxiliary sets.

Sources

Source comparison: Rouquier §3.2.1-3.2.4 (presentation, Cartan specialization, quiver-with-automorphism formula); Khovanov–Lauda I §2.1-2.2 and Khovanov–Lauda II §1-§2 (diagrammatic presentation with the dot counts $d_{i j}$ ); Kleshchev §2.11 Definition 2.14 (the affine presentation). The printed Rouquier sign in relation (6) is the opposite of the one fixed here; the passage is $τ_{a} \mapsto - τ_{a}$ as recorded above.

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

The KLR grading and the diagrammatic multiplication

Definition

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum. Assign degrees $\deg e (ν) : = 0$ , $\deg (x_{a} e (ν)) : = (α_{ν_{a}} ∣ α_{ν_{a}}) = 2 d_{ν_{a}}$ and $\deg (τ_{a} e (ν)) : = - (α_{ν_{a}} ∣ α_{ν_{a + 1}}) = - b_{ν_{a} ν_{a + 1}}$ in the presentation; then every relation (1)-(7) is homogeneous (## Verification), so $R_{n} (Q) = ⨁_{m \in Z} R_{n} (Q)_{m}$ is a $Z$ -graded unital associative $k$ -algebra with $R_{n} (Q)_{m} = ⨁_{ν, η \in I^{n}} e (ν) R_{n} (Q)_{m} e (η)$ , and each $R (β) = ⨁_{m} R (β)_{m}$ is a graded subalgebra (Associative graded algebras, bimodules, and internal shifts). The global sums $x_{a} = \sum_{ν} x_{a} e (ν)$ and $τ_{a} = \sum_{ν} τ_{a} e (ν)$ need not be homogeneous; displayed generator degrees always refer to their word components. Equivalently, the weight refinement by $Q_{+}$ is available: the idempotent $e (ν)$ carries weight $\sum_{a} α_{ν_{a}}$ , and $x_{a}$ and $τ_{a}$ are weight-preserving operators of the displayed scalar degrees.

Diagram dictionary. Read a planar diagram bottom to top. A local dot on strand $a$ is $x_{a}$ , a crossing of strands $a, a + 1$ is $τ_{a}$ , the identity is the collection of straight diagrams $e (ν)$ , and vertical concatenation of diagrams with matching bottom labels $ν$ and top labels $s_{a_{r}} \dots s_{a_{1}} ν$ is the algebra product: in $a b$ the diagram for $b$ is placed below the diagram for $a$ , matching operator composition. Isotopies of the plane that do not change the combinatorial type act as the identity, and the local relations (2.3)-(2.8) of Khovanov–Lauda I (quadratic relation, dot-sliding over a crossing, and the braid relation) in the simply-laced Khovanov–Lauda parameter specialization are relations (4), (6) and (7); for general $Q$ the local diagram relations are defined by the full presentation of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum. In particular the degree of a monomial equals the sum of the individual dot degrees plus the signed sum of the crossing degrees, and a dot contributes $(α_{ν_{a}} ∣ α_{ν_{a}}) = 2 d_{ν_{a}}$ , while a crossing contributes $- b_{ν_{a} ν_{a + 1}}$ .

All grading shifts on this page are the internal shifts of Associative graded algebras, bimodules, and internal shifts, $(M {r})_{m} = M_{m - r}$ , and $q$ acts on graded Grothendieck groups by $q [P] = [P {1}]$ .

Facts & Assumptions

Given: A KLR datum $Q$ and the presented algebra $R_{n} (Q)$ of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum with its relations (1)-(7), together with the degree assignment $\deg e (ν) = 0$ , $\deg x_{a} = 2 d_{ν_{a}}$ , $\deg τ_{a} = - b_{ν_{a} ν_{a + 1}}$ and the symmetric matrix $b_{i j} = d_{i} a_{i j}$ with $b_{i i} = 2 d_{i} > 0$ of Kac Moody root lattice height and positive cone.

[F1]

The relation list of the KLR algebra: relations (1)-(7) of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum, including the symmetry $b_{i j} = b_{j i}$ and $b_{i i} = 2 d_{i}$ .

[F2]

The parameter conditions (Q1)-(Q4) of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum, in particular (Q3): $Q_{i j}$ is homogeneous of total degree $- 2 b_{i j}$ for $\deg u = 2 d_{i}$ , $\deg v = 2 d_{j}$ .

Verification

technique · direct; check homogeneity of each defining relation for the displayed degree assignment and then transfer it to the quotient

1.1givenF1algebra

Relation (1) has degree $0$ on both sides, and in (2) the identity $x_{a} e (ν) = e (ν) x_{a}$ has degree $2 d_{ν_{a}}$ while $x_{a} x_{b} = x_{b} x_{a}$ has degree $2 d_{ν_{a}} + 2 d_{ν_{b}}$ on the $ν$ -component; relation (3) has degree $- b_{ν_{a} ν_{a + 1}} = - b_{ν_{a + 1} ν_{a}}$ on both sides by the symmetry of $b$ from [F1].

2.1step 1.1F1F2algebra

Relation (4) is homogeneous of degree $- 2 b_{ν_{a} ν_{a + 1}}$ : the left side $τ_{a}^{2} e (ν)$ has degree $2 (- b_{ν_{a} ν_{a + 1}})$ , and the right side $Q_{ν_{a} ν_{a + 1}} (x_{a}, x_{a + 1}) e (ν)$ has degree $- 2 b_{ν_{a} ν_{a + 1}}$ by [F2], since a monomial $u^{r} v^{s}$ of $Q_{i j}$ contributes $r (2 d_{i}) + s (2 d_{j})$ .

2.2step 1.1F1algebra

Relations (5) and (6) are homogeneous with their actual label-dependent degrees. In far crossing commutation, both products have degree $- b_{ν_{a} ν_{a + 1}} - b_{ν_{b} ν_{b + 1}}$ because the disjoint crossings interchange the same two color pairs. A far dot move has degree $2 d_{ν_{b}} - b_{ν_{a} ν_{a + 1}}$ on both sides, as its untouched strand keeps the dot color. For adjacent unequal colors, dot sliding likewise preserves the degree of the crossing plus the transported dot. For equal colors $i$ , the dot degree is $2 d_{i}$ and the crossing degree is $- 2 d_{i}$ , so both terms of $τ_{a} x_{a} - x_{a + 1} τ_{a} = e (ν)$ and $τ_{a} x_{a + 1} - x_{a} τ_{a} = - e (ν)$ have degree $0$ , equal to the degree of the idempotent on the right. Thus every dot and far-crossing relation is homogeneous, without asserting that its common degree is always zero.

3.1step 2.2F1F2algebra

Relation (7), unequal outer colours: for colours $(ν_{a}, ν_{a + 1}, ν_{a + 2}) = (a^{'}, b^{'}, c^{'})$ both braid words are sums of terms each moving the same three crossings over the same three pairs, so each has degree $- b_{a^{'} b^{'}} - b_{a^{'} c^{'}} - b_{b^{'} c^{'}}$ , and the right side is $0$ ; on the equal-outer-colour case $(i, j, i)$ each braid word has the same degree $- 2 b_{i j} - 2 d_{i}$ , while the correction numerator has degree $- 2 b_{i j}$ and its division by $x_{a + 2} - x_{a}$ , of degree $2 d_{i}$ , lowers the degree to $- 2 b_{i j} - 2 d_{i}$ ; for three equal colours $Q_{i i} = 0$ and the ordinary nil-Hecke braid relation is homogeneous of degree $- 6 d_{i}$ .

4.1step 1.1step 2.1step 2.2step 3.1algebra∎

Use the free path algebra with vertices $ν$ , loops $x_{a} e (ν)$ and arrows $τ_{a} e (ν) : ν \to s_{a} ν$ , assigned the displayed degrees. Its vertex sum is the unit because $I$ is finite; decomposing the global generators by word idempotents identifies its presented quotient with $R_{n} (Q)$ . Every defining component relation is homogeneous, so the two-sided ideal they generate is graded, and the quotient $R_{n} (Q)$ inherits a $Z$ -grading with $R_{n} (Q)_{m} = ⨁_{ν, η} e (ν) R_{n} (Q)_{m} e (η)$ ; the weight refinement is inherited because each generator preserves the weight $\sum_{a} α_{ν_{a}}$ and each relation is weight-homogeneous, and $R (β) = e (β) R_{∣ β ∣} (Q) e (β)$ is a graded subalgebra for every $β \in Q_{+}$ .

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Faithful polynomial representation of a KLR algebra

Statement

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum. Choose a factorization $Q_{i j} (u, v) = P_{i j} (u, v) P_{j i} (v, u)$ with $P_{i j} \in k [u, v]$ nonzero for all $i \neq j$ ; with (Q1) one may order the pairs and take $P_{i j} = Q_{i j}$ , $P_{j i} = 1$ . On the graded space ${P o l}_{n} : = ⨁_{ν \in I^{n}} k [x_{1}, \dots, x_{n}] e (ν)$ define operators: $e (ν)$ projects to the $ν$ -summand; $x_{a}$ multiplies by $x_{a}$ ; for $ν_{a} \neq ν_{a + 1}$ set $τ_{a} (f e (ν)) : = P_{ν_{a} ν_{a + 1}} (x_{a + 1}, x_{a}) s_{a} (f) e (s_{a} ν)$ where $s_{a}$ permutes the variables; for $ν_{a} = ν_{a + 1}$ set $τ_{a} (f e (ν)) : = \frac{f - s_{a} (f)}{x_{a} - x_{a + 1}} e (ν)$ , the divided difference operator of the sign convention fixed in relation (6) of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum (the quotient lies in $k [x_{1}, \dots, x_{n}]$ because $f - s_{a} f$ vanishes at $x_{a} = x_{a + 1}$ ; the variable permutation of Polynomial rings in finitely many commuting indeterminates by iteration makes this precise). Give ${P o l}_{n}$ the grading of The KLR grading and the diagrammatic multiplication with $\deg x_{a} e (ν) : = 2 d_{ν_{a}}$ , together with the summand shifts that make each $τ_{a} e (ν)$ homogeneous of degree $- b_{ν_{a} ν_{a + 1}}$ (they exist by the antisymmetry of the increments $- b_{i j} - \deg P_{i j}$ and are unique up to one constant on each fixed-weight sequence orbit). Then each word-component generator acts homogeneously of its degree, the operators satisfy relations (1)-(7), and they define a homomorphism of graded algebras $ρ : R_{n} (Q) \to {E n d}_{k}^{g r} ({P o l}_{n})$ , the graded algebra of homogeneous endomorphisms, with $ρ (R_{n} (Q)_{m}) {P o l}_{n, d} \subseteq {P o l}_{n, d + m}$ . For nonzero $P_{i j}$ this homomorphism is injective. After localizing at the nonzero polynomials $P_{i j} (x_{a}, x_{b})$ ( $ν_{a} \neq ν_{b}$ ) and the differences $x_{a} - x_{b}$ ( $ν_{a} = ν_{b}$ ), it becomes an isomorphism onto the localization of the skew group algebra $A_{n} (I) = (⨁_{ν \in I^{n}} k [x_{1}, \dots, x_{n}] e (ν)) ⋊ S_{n}$ ; consequently $R_{n} (Q)$ has finite-dimensional homogeneous components and $ρ$ is a faithful representation.

Facts & Assumptions

Given: A KLR datum $Q$ of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum with $b_{i j} = d_{i} a_{i j} = b_{j i}$ and $b_{i i} = 2 d_{i} > 0$ , a factorization $Q_{i j} (u, v) = P_{i j} (u, v) P_{j i} (v, u)$ with all $P_{i j} \neq 0$ , and the space ${P o l}_{n} = ⨁_{ν \in I^{n}} k [x_{1}, \dots, x_{n}] e (ν)$ with the degree assignment $\deg x_{a} = 2 d_{ν_{a}}$ and idempotents of degree~ $0$ .

[F1]

(Q3) of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum: $Q_{i j}$ is homogeneous of total degree $- 2 b_{i j}$ for $\deg u = 2 d_{i}$ , $\deg v = 2 d_{j}$ .

[F2]

Homogeneity of factors: for the positive weighting $[i, j]$ with $\deg u = 2 d_{i}$ , $\deg v = 2 d_{j}$ , the least and greatest weighted degrees of a product of nonzero polynomials are the sums of those of its factors, since the polynomial ring over $k$ is a domain. Thus (Q3) and $Q_{i j} = P_{i j} (u, v) P_{j i} (v, u)$ force both factors to be homogeneous, with $\deg_{[i, j]} P_{i j} + \deg_{[j, i]} P_{j i} = - 2 b_{i j}$ . No homogeneity for the swapped weighting is asserted.

[F3]

Relation list (1)-(7) of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum, and the identity for divided differences: with $s_{a}$ acting on polynomials by $x_{a} \leftrightarrow x_{a + 1}$ and $\partial_{a} = (1 - s_{a}) / (x_{a} - x_{a + 1})$ , one has $\partial_{a} x_{a} - x_{a + 1} \partial_{a} = 1$ , $\partial_{a} x_{a + 1} - x_{a} \partial_{a} = - 1$ , $\partial_{a}^{2} = 0$ and $\partial_{a} (f g) = \partial_{a} (f) g + s_{a} (f) \partial_{a} (g)$ for all polynomials $f, g$ .

[F4]

Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups, clauses (1)–(2), connects reduced expressions by braid moves and transforms a nonreduced word by braid moves to an adjacent repeat; here the Coxeter system is that of adjacent transpositions in $S_{n}$ .

Proof

technique · direct; construct the shifts, verify the relations case by case, then prove injectivity and the localized isomorphism

1.1givenF2algebra

(Shifts.) Write $p_{i j} : = \deg_{[i, j]} P_{i j}$ and $Δ_{i j} : = - b_{i j} - p_{i j}$ for $i \neq j$ . By [F2], $p_{i j} + p_{j i} = - 2 b_{i j}$ , so $Δ_{j i} = - Δ_{i j}$ . Fix the given finite order on $I$ and set $w (ν) : = \sum_{a < b, ν_{a} > ν_{b}} Δ_{ν_{b} ν_{a}}$ . An adjacent unequal swap only changes the contribution of that pair, hence $w (s_{a} ν) - w (ν) = Δ_{ν_{a} ν_{a + 1}}$ . This explicit formula supplies the shifts on every sequence; any two shift functions differ by a constant on each fixed-content orbit, since adjacent unequal swaps connect it.

1.2F3F4algebrainduction

(Spanning.) Move dots to the right using relation (6), inducting on the number of dot/crossing inversions and the crossing length of correction terms. By [F4], any nonreduced transposition word admits finitely many braid moves exposing an adjacent repeat, and any reduced word connects to the chosen one by braid moves. Each braid correction has three fewer crossings and each quadratic substitution removes two; induction on crossing length therefore reduces all correction terms, while the finite chosen sequence of braid moves handles the leading term. Hence the monomials $τ_{w} x^{a} e (ν)$ span.

2.1givenF1F2algebraconstruct

(Definition and homogeneity.) Define $e (ν)$ and $x_{a}$ as stated and $τ_{a}$ by the two displayed formulas. Each formula is a finite sum of polynomials times a permutation of the variables, hence a well-defined $k$ -linear endomorphism; the equal-colour quotient lies in $k [x_{1}, \dots, x_{n}]$ because $f - s_{a} f$ vanishes on $x_{a} = x_{a + 1}$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, Polynomial rings in finitely many commuting indeterminates by iteration). With the shifts of step 1.1, on each word component $x_{a}$ is homogeneous of degree $2 d_{ν_{a}}$ (the shifts are constant on a summand and do not change dot degrees), and $τ_{a}$ is homogeneous of degree $- b_{ν_{a} ν_{a + 1}}$ : on unequal colours in the target summand $s_{a} ν$ , the variable $x_{a + 1}$ has color $p$ and $x_{a}$ has color $q$ , so the factor $P_{p q} (x_{a + 1}, x_{a})$ has degree $\deg_{[p, q]} P_{p q}$ , so multiplying by it and passing to the summand $s_{a} ν$ with shift $w (s_{a} ν) = w (ν) + Δ_{p q}$ changes the total degree by $\deg_{[p, q]} P_{p q} + Δ_{p q} = - b_{p q}$ ; on equal colours the divided difference has degree $- 2 d_{p} = - b_{p p}$ because its numerator has degree $\deg f$ and the denominator has degree $2 d_{p}$ .

3.1step 2.1F3algebra

(Relations (1), (2), (3), (5), (6).) Relations (1)-(3) are immediate from the definitions: the $e (ν)$ are orthogonal idempotents summing to $1$ , $x$ 's commute and commute with the $e (ν)$ , and both displayed $τ_{a}$ 's map the $ν$ -summand to the $s_{a} ν$ -summand. Relation (5) holds because $τ_{a}$ and $τ_{b}$ with $∣ a - b ∣ > 1$ act on disjoint sets of variables and their polynomial factors commute. For (6): if $b \notin {a, a + 1}$ the multiplication by $x_{b}$ commutes with $s_{a}$ and with the factor $P_{ν_{a} ν_{a + 1}} (x_{a + 1}, x_{a})$ , giving $τ_{a} x_{b} = x_{b} τ_{a} = x_{s_{a} (b)} τ_{a}$ ; if $b = a$ and $ν_{a} \neq ν_{a + 1}$ then $τ_{a} x_{a} f = P (x_{a + 1}, x_{a}) x_{a + 1} s_{a} f = x_{a + 1} τ_{a} f$ , and $b = a + 1$ is the same with $x_{a}$ ; if $ν_{a} = ν_{a + 1}$ the two identities $\partial_{a} x_{a} - x_{a + 1} \partial_{a} = 1$ and $\partial_{a} x_{a + 1} - x_{a} \partial_{a} = - 1$ of [F3] are exactly the two displayed dot moves.

3.2step 2.1F1F3algebra

(Relation (4).) On equal colours $τ_{a}^{2} = \partial_{a}^{2} = 0 = Q_{p p} (x_{a}, x_{a + 1}) e (ν)$ by [F3] and $Q_{p p} = 0$ . On unequal colours $(p, q)$ a double application gives $τ_{a}^{2} (f e (ν)) = P_{q p} (x_{a + 1}, x_{a}) s_{a} (P_{p q} (x_{a + 1}, x_{a}) s_{a} f) = P_{p q} (x_{a}, x_{a + 1}) P_{q p} (x_{a + 1}, x_{a}) f = Q_{p q} (x_{a}, x_{a + 1}) f$ , which is relation (4) because $Q_{p q} (u, v) = P_{p q} (u, v) P_{q p} (v, u)$ evaluated at $(u, v) = (x_{a}, x_{a + 1})$ equals $P_{p q} (x_{a}, x_{a + 1}) P_{q p} (x_{a + 1}, x_{a})$ .

4.1step 2.1step 3.2F3algebra

(Relation (7), equal outer colours $(i, j, i)$ .) Work on $e (i, j, i)$ in variables $x_{1}, x_{2}, x_{3} = x_{a}, x_{a + 1}, x_{a + 2}$ , write $P = P_{i j}$ , $P^{'} = P_{j i}$ , $Q = P P^{'} (v, u)$ , let $s = (12)$ , $t = (23)$ and $r = t s t = s t s = (13)$ ; write $A = τ_{a + 1} τ_{a} τ_{a + 1}$ and $B = τ_{a} τ_{a + 1} τ_{a}$ . Expanding the definitions on $f e (i, j, i)$ and collecting the outer factor, $A f = P (x_{3}, x_{2}) \cdot t [\partial_{a} (P^{'} (x_{3}, x_{2}) t f)]$ and $B f = P^{'} (x_{2}, x_{1}) \cdot s [\partial_{a + 1} (P (x_{2}, x_{1}) s f)]$ . Using $t \partial_{a} t = (1 - r) / (x_{1} - x_{3})$ and $\partial_{a + 1} (h g) = \partial_{a + 1} (h) g + t (h) \partial_{a + 1} (g)$ with $h = P^{'} (x_{3}, x_{2})$ first, then the mirrored expansion of $B$ with $h = P (x_{2}, x_{1})$ , gives $A f = P (x_{3}, x_{2}) [P^{'} (x_{2}, x_{3}) f - P^{'} (x_{2}, x_{1}) r f] / (x_{1} - x_{3})$ and $B f = P^{'} (x_{2}, x_{1}) [P (x_{1}, x_{2}) f - P (x_{3}, x_{2}) r f] / (x_{1} - x_{3})$ . Subtracting, the two $r f$ -terms cancel and the remaining coefficient is $P (x_{3}, x_{2}) P^{'} (x_{2}, x_{3}) - P (x_{1}, x_{2}) P^{'} (x_{2}, x_{1}) = Q (x_{3}, x_{2}) - Q (x_{1}, x_{2})$ , so $(A - B) f = \frac{Q (x_{1}, x_{2}) - Q (x_{3}, x_{2})}{x_{3} - x_{1}} f$ , which is relation (7) on $e (i, j, i)$ .

5.1step 3.1step 4.1F1F3algebra

(Relation (7), remaining colours.) If $ν_{a} = ν_{a + 1} = ν_{a + 2} = i$ , then all three operators are divided differences and both braid words equal the same antisymmetrizer expression $\sum_{w \in S_{3}} sgn (w) w / \prod_{i < j} (x_{i} - x_{j})$ , so $A - B = 0$ , consistent with $Q_{i i} = 0$ ; if exactly the outer two colours agree, step 4.1 applies with the two adjacent colours distinct; if only the first two colours agree, say $(i, i, j)$ , then with $h = P_{i j} (x_{2}, x_{1}) P_{i j} (x_{3}, x_{1})$ , which is symmetric in $x_{2}, x_{3}$ , the two expansions give $A f = \partial_{a + 1} (h \cdot s t f)$ and $B f = h \cdot s t \partial_{a} (f)$ , and the product rule plus the Demazure identity $\partial_{a + 1} s t = s t \partial_{a}$ valid for the $3$ -cycle $w = s t$ yield $A - B = 0$ ; the case $(i, j, j)$ is the mirror image, obtained by reading the same identities from the other end. If the three colours are distinct, both $A$ and $B$ act as the same product $P_{i j} P_{i k} P_{j k}$ (with arguments placed by the two expansions, all three factors read off the three crossed pairs) times the longest permutation $r$ , and $A = B$ ; the expansion is by the same substitution $t \partial_{a} t = (1 - r) / (x_{1} - x_{3})$ as in step 4.1 applied to three distinct colours, where the two middle expressions coincide because the two factorizations of the long element have the same length. Hence relation (7) holds in every case.

6.1step 2.1step 3.1step 3.2step 4.1step 5.1construct

(The homomorphism.) Steps 3.1, 3.2, 4.1 and 5.1 verify relations (1)-(7) for the operators, and step 2.1 shows each word-component generator acts homogeneously of its assigned degree, so the universal property of the presented algebra gives a homomorphism $ρ : R_{n} (Q) \to {E n d}_{k}^{g r} ({P o l}_{n} {w})$ of graded algebras with $ρ (R_{n} (Q)_{m}) {P o l}_{n, d} \subseteq {P o l}_{n, d + m}$ .

7.1step 6.1step 1.2choosealgebra

(Faithfulness.) Localize for this step only at the nonzero elements ${P_{i j} (x_{a}, x_{b}) : ν_{a} \neq ν_{b}}$ and ${x_{a} - x_{b} : ν_{a} = ν_{b}}$ ; the localized operators of a reduced word $w = s_{i_{1}} \dots s_{i_{r}}$ on a fixed source sequence $ν$ have the form $c_{w} \cdot w + \sum_{u < w} c_{u} u$ where $c_{w} \neq 0$ is the product of the nonzero $P$ -factors (unequal crossings) and the nonzero coefficient $- 1 / (x_{a} - x_{a + 1})$ (equal crossings), and the terms $u < w$ are strictly shorter permutations. Suppose $\sum_{ν} \sum_{w, a} c_{w, a} τ_{w} x^{a} e (ν)$ acted as zero; for the maximal $w$ occurring in a fixed source/destination block the leading terms give a nontrivial linear relation $\sum_{w} d_{w} w = 0$ among distinct automorphisms $w$ of the field $k (x_{1}, \dots, x_{n})$ . Distinct automorphisms are linearly independent: take a relation with the least positive number of nonzero coefficients; one term cannot vanish on $1$ . With at least two terms, choose $w \neq w_{0}$ and $g$ with $w (g) \neq w_{0} (g)$ , evaluate on $f g$ , and subtract $w_{0} (g)$ times the evaluation on $f$ . The $w_{0}$ term disappears and the $w$ coefficient stays nonzero, contradicting minimality; hence no nonzero combination of the spanning monomials of step 1.2 acts as zero and $ρ$ is injective.

8.1step 1.2step 7.1algebra∎

(Localized isomorphism and finiteness.) Inverting the factors and differences makes every unequal-colour crossing $τ_{a}$ equal to $s_{a}$ up to an invertible scalar, and every equal-colour crossing satisfy $s_{a} = 1 - (x_{a} - x_{a + 1}) \partial_{a}$ , so the localized image contains all variable permutations $s_{a}$ and all multiplications $x_{a}$ , i.e. the localization of the skew group algebra $A_{n} (I) = (⨁_{ν \in I^{n}} k [x_{1}, \dots, x_{n}] e (ν)) ⋊ S_{n}$ ; with injectivity from step 7.1 and the evident containment of the image in the localized algebra, $ρ$ becomes an isomorphism after localization. For finiteness of homogeneous components: by step 1.2 the monomials with $w \in S_{n}$ and dot exponents $a_{j} \geq 0$ span $R_{n} (Q)$ ; for fixed weight $β$ and degree $m$ the finitely many $w$ and $ν$ occur, each crossing degree is one of finitely many integers bounded below, and each dot has positive degree $2 d_{ν_{a}}$ , so the total dot degree of a spanning monomial of degree $m$ is bounded, leaving only finitely many exponent tuples; hence $\dim_{k} R (β)_{m} < \infty$ .

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The PBW (monomial) basis theorem for KLR algebras

Statement

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum and fix a reduced-expression system: for each $w \in S_{n}$ one reduced word $w = s_{i_{1}} \dots s_{i_{r}}$ , and for $ν \in I^{n}$ put $τ_{w} : = τ_{i_{1}} \dots τ_{i_{r}} \in R_{n} (Q)$ . Then the monomials $τ_{w} x_{1}^{a_{1}} \dots x_{n}^{a_{n}} e (ν)$ , over all $w \in S_{n}$ , $a_{1}, \dots, a_{n} \geq 0$ and $ν \in I^{n}$ , form a $k$ -basis of $R_{n} (Q)$ ; equivalently $R_{n} (Q)$ is a free $k$ -module on these monomials, and $R (β)$ has the corresponding basis. Moreover each $R (β)$ has finite-dimensional homogeneous components, and the graded dimension of the summand $e (ν) R_{n} (Q) e (ν^{'})$ is the formal Laurent Hilbert series counting the monomials (bounded below, equivalently the finite sum of the corresponding rational geometric series), so the graded ranks are independent of the parameters $Q$ : for each fixed $β$ , $\dim_{k} R (β)_{m} < \infty$ for all $m$ and $R (β)_{m} = 0$ for $m ≪ 0$ ; if $I$ is finite the same holds for $R_{n} (Q)$ . In particular $R (β)$ is a locally finite graded algebra; its degree-zero part need not be one-dimensional (already $x_{1} τ_{1} \in R (2 α_{i})_{0}$ is a nontrivial idempotent).

Facts & Assumptions

Given: A KLR datum $Q$ with the algebra $R_{n} (Q)$ of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum, its grading of The KLR grading and the diagrammatic multiplication, a reduced-expression system for $S_{n}$ , and the faithful graded polynomial representation of Faithful polynomial representation of a KLR algebra on ${P o l}_{n}$ .

[F1]

Faithfulness (Faithful polynomial representation of a KLR algebra): the operators of the polynomial representation satisfy relations (1)-(7) and the homomorphism $ρ : R_{n} (Q) \to {E n d}_{k}^{g r} ({P o l}_{n} {w})$ is injective for nonzero $P_{i j}$ , and it becomes an isomorphism onto the localized skew group algebra $k (I) [x] ⋊ S_{n}$ after inverting the nonzero polynomials $P_{i j} (x_{a}, x_{b})$ and differences $x_{a} - x_{b}$ .

[F2]

Relations (4), (5), (6), (7) of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum, used as rewriting rules for words in the generators, and the degree assignment $\deg x_{a} = 2 d_{ν_{a}} > 0$ , $\deg τ_{a} = - b_{ν_{a} ν_{a + 1}}$ of The KLR grading and the diagrammatic multiplication.

[F3]

Reduced-word moves and deletion: Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups, clauses (1) and (2), applied to the type- $A$ Coxeter system of adjacent transpositions, connects reduced words by far and three-term braid moves and transforms any nonreduced word by such moves to an adjacent repeat.

Proof

technique · direct; first reduce arbitrary words to monomials with reduced crossing words and right-hanging dots, then obtain independence from the faithful polynomial representation

1.1givenF2algebra

(Dots move right.) Move the word idempotents to the right using relations (2)–(3). On each resulting component, if $x_{b}$ stands immediately left of $τ_{a}$ , relation (6) gives $x_{b} τ_{a} e (ν) = τ_{a} x_{s_{a} (b)} e (ν)$ except for adjacent equal colours. In that case the exact rules are $x_{a} τ_{a} e (ν) = τ_{a} x_{a + 1} e (ν) + e (ν)$ and $x_{a + 1} τ_{a} e (ν) = τ_{a} x_{a} e (ν) - e (ν)$ . The leading term moves one dot past one crossing, and each correction removes that dot and crossing; hence every term has fewer dot/crossing inversions. Induction on their number puts all dots to the right of all crossings.

2.1step 1.1F2F3inductionalgebra

(Crossing words reduce.) Induct on crossing length. By [F3], a nonreduced transposition word can be changed by finitely many far and braid moves into a word with an adjacent repeat; each braid correction has three fewer crossings and is handled by induction after step 1.1, and relation (4) removes the repeated pair. For a reduced word, [F3] connects it to the chosen reduced word by finitely many braid moves; their correction terms again have fewer crossings. Thus every crossing word is a sum of chosen $τ_{w}$ times dot polynomials, and step 1.1 puts the dots on the right. This proves spanning without a decreasing-length claim for braid moves themselves.

3.1step 2.1F1choosealgebra

(Independence.) Under the representation of [F1], the image of $τ_{w} x^{a} e (ν)$ in the localization at the nonzero $P_{i j} (x_{a}, x_{b})$ and $x_{a} - x_{b}$ is triangular: expanding the operators of the chosen reduced word, its highest term is a nonzero scalar times the permutation $w$ acting on the summand, and all other terms are permutations of strictly smaller length (an unequal crossing contributes its nonzero $P$ -factor together with the transposition, an equal crossing contributes $(1 - s_{a}) / (x_{a} - x_{a + 1}) = - (s_{a} - 1) / (x_{a} - x_{a + 1})$ , again transposition plus lower-length terms). A linear relation $\sum c_{w, a, ν} τ_{w} x^{a} e (ν) = 0$ in $R_{n} (Q)$ would therefore give, after applying $ρ$ and looking at a source/destination block and the maximal occurring length, a nontrivial linear relation $\sum_{w} d_{w} w = 0$ among distinct permutations $w$ of the field $k (x_{1}, \dots, x_{n})$ . Distinct field automorphisms are linearly independent: in a shortest nonzero relation choose two different automorphisms $w, w_{0}$ and $g$ with $w (g) \neq w_{0} (g)$ . Evaluation on $f g$ minus $w_{0} (g)$ times evaluation on $f$ removes one term and leaves the $w$ term nonzero, contradicting minimality; a one-term relation is impossible by evaluation on $1$ . Hence the spanning monomials of step 2.1 have linearly independent images, and since $ρ$ is injective by [F1] applied to the fixed factorization, the monomials themselves are linearly independent; together with step 2.1 they form a basis.

4.1step 3.1F2algebra

(Grading bookkeeping.) The bidegree of the basis monomial $τ_{w} x_{1}^{a_{1}} \dots x_{n}^{a_{n}} e (ν)$ is $(ν, w ν)$ together with the integer $\deg (τ_{w} on ν) + \sum_{j} a_{j} \cdot 2 d_{ν_{j}}$ , where $\deg (τ_{w} on ν)$ is the sum of the degrees $- b_{ν_{r} ν_{r + 1}}$ of its crossings read in the successive permuted sequences; this is independent of the parameters $Q$ and of the factorization. The summand $e (ν) R_{n} (Q) e (ν^{'})$ therefore has the Laurent Hilbert series $\sum_{w : w ν^{'} = ν} q^{\deg (τ_{w} on ν^{'})} \prod_{j = 1}^{n} (1 - q^{2 d_{ν_{j}^{'}}})^{- 1}$ , a finite sum of rational geometric series bounded below (each $d_{ν_{j}} > 0$ and the finitely many crossing degrees are bounded below), which proves the displayed graded-dimension claim and shows that $R (β)_{m} = 0$ for $m$ below the finite minimum of the finitely many crossing contributions over the finitely many $ν, w$ with $w ν = ν^{'}$ and $\sum_{j} α_{ν_{j}} = β$ .

5.1step 3.1step 4.1algebra∎

(Local finiteness.) Fix $β \in Q_{+}$ and a degree $m$ . The basis monomials of $R (β)_{m}$ involve only the finitely many sequences $ν \in I^{β}$ and the finitely many $w \in S_{∣ β ∣}$ ; each crossing degree is one of the finitely many integers $- b_{p q}$ and each dot degree $2 d_{ν_{j}}$ is at least $2 \min_{i} d_{i} > 0$ , so $\sum_{j} a_{j}$ is bounded, leaving finitely many exponent tuples; hence $\dim_{k} R (β)_{m} < \infty$ . If $I$ is finite, $I^{β}$ is finite for every $∣ β ∣$ , so the same argument applies to $R_{n} (Q)$ degree by degree. The element $x_{1} τ_{1} \in R (2 α_{i})_{0}$ is nonzero by the basis of step 3.1 and squares to itself by the nil-Hecke dot move, so degree zero need not be one-dimensional.

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Induction and restriction bimodules for KLR algebras

Definition

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum. For $β, γ \in Q_{+}$ put $e (β, γ) : = \sum_{ν \in I^{β}, ν^{'} \in I^{γ}} e (ν * ν^{'}) \in R (β + γ)$ , where $ν * ν^{'}$ is concatenation; this is an idempotent. The block inclusions $R (β) ↪ R (β + γ)$ , $r \mapsto r \otimes 1$ and $R (γ) ↪ R (β + γ)$ , $s \mapsto 1 \otimes s$ combine into an injective $k$ -algebra map $R (β) \otimes_{k} R (γ) \to R (β + γ)$ whose image is the separated-block subalgebra of the corner $e (β, γ) R (β + γ) e (β, γ)$ ; the inclusion generally is proper; in the diagram calculus it is horizontal concatenation of diagrams. Induction and restriction are the functors on graded left modules ${I n d}_{β, γ} (M ⊠ N) : = R (β + γ) e (β, γ) \otimes_{R (β) \otimes_{k} R (γ)} (M \otimes_{k} N), {R e s}_{β, γ} L : = e (β, γ) L,$ where $\otimes_{R (β) \otimes_{k} R (γ)}$ is the graded balanced tensor product of Graded balanced tensor product and homogeneous Hom (total-degree grading) and $M \otimes_{k} N$ carries the graded tensor product structure; $I n d$ is a functor on graded modules over the graded algebra $R (β) \otimes_{k} R (γ)$ (through the tensor product of the two graded modules) and, since $R (β + γ) e (β, γ)$ is a graded $(R (β + γ), R (β) \otimes_{k} R (γ))$ -bimodule and $e (β, γ) R (β + γ)$ is its right-module counterpart, the two tensor functors are of the two sidedness types of Associative and graded bimodule tensor–Hom adjunction. Both functors are additive and degree-zero on degree-zero maps and extend to the direct sums over weights. Shifts satisfy ${I n d}_{β, γ} (M {r} ⊠ N {s}) ≅ {I n d}_{β, γ} (M ⊠ N) {r + s}$ and ${R e s}_{β, γ} (L {r}) ≅ ({R e s}_{β, γ} L) {r}$ (Graded associativity, units, and internal-shift tensor isomorphisms). For a partial flag $β = β_{1} + \dots + β_{r}$ the iterated functors are defined by iteration, and the parenthesization-independent identification is the one consumed below.

Remarks

Well-definedness. Concatenation sends the two tensor-factor identities to $e (β, γ)$ and respects the relations of each block, so $r \otimes s \mapsto r \cdot (image of s)$ is a unital algebra homomorphism from $R (β) \otimes_{k} R (γ)$ onto the separated-block subalgebra of the corner; note that it is unital into the corner, not into all of $R (β + γ)$ , and the two block images commute because separated generators commute by relations (3), (5) and (6) with $∣ a - b ∣ > 1$ . Injectivity is read off the PBW basis of The PBW (monomial) basis theorem for KLR algebras: the monomials whose permutation word lies in the parabolic subgroup $S_{∣ β ∣} \times S_{∣ γ ∣}$ (in the appropriate block positions) and whose dot exponents are arbitrary are exactly the images of the tensor products of the two PBW bases, hence are linearly independent. The image can be strictly smaller than the corner: for $β = γ = α_{i}$ the corner is the nil-Hecke algebra $R (2 α_{i}) ≅ N H_{2}$ , while the image is $k [x_{1}, x_{2}] e (i i)$ (with $x_{1}, x_{2}$ acting blockwise), which does not contain $τ_{1}$ .

Functoriality and shifts. The balanced tensor product of a right and a left module over the graded algebra $R (β) \otimes_{k} R (γ)$ is defined by the universal property with respect to balanced bilinear maps of degree $0$ , and both balance relations and the outer $R (β + γ)$ -action preserve degrees; moving an internal shift through the total-degree tensor grading gives the two displayed shift isomorphisms. For an iterated flag, the maps $a \otimes (b \otimes m) \mapsto a b \otimes m$ identify the balanced tensors of the different parenthesizations, so the iteration is defined up to canonical degree-zero isomorphism; the classes in the Grothendieck groups below are unaffected. The construction is choice-free: no representatives or auxiliary data are selected.

Sources

Source comparison: Khovanov–Lauda I §2.6 (the embedding, the idempotent $1_{ν, ν^{'}}$ and the functors); Kang–Kashiwara §3 (3.9)-(3.11) (the $R (m) \otimes R (n)$ -balanced form used here); Kleshchev §2.11 (multi-factor embeddings and the induced/restricted functors).

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Generic KLR induction and restriction adjunctions preserve projectives

Statement

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum and $β, γ \in Q_{+}$ . Put $A = R (β + γ)$ , $B = R (β) \otimes_{k} R (γ)$ , $e = e (β, γ)$ and $c = (β ∣ γ)$ . The bimodules $e A$ and $A e$ are ungraded free respectively as left and right $B$ -modules, of rank $(\binom{∣ β ∣ + ∣ γ ∣}{∣ β ∣})$ . As GRADED modules they are finite projective direct sums of the appropriate principal idempotent $B$ -modules with labelled shuffle shifts: for each concatenated word $ξ = ν * η$ the right-module summand $e (ξ) B$ has shift $δ_{R} (w, ξ) = \deg (τ_{w} e (ξ))$ and the left-module summand $B e (ξ)$ has shift $δ_{L} (w, ξ) = \deg (e (ξ) τ_{w}) = δ_{R} (w, w^{- 1} ξ)$ , using respectively the corresponding right and left minimal shuffle representatives $w$ ; these degrees depend on the labels, and no common shift for an entire $B$ copy is asserted. The functors $I n d = A e \otimes_{B} -$ and $R e s = e (-)$ of Induction and restriction bimodules for KLR algebras are exact, commute with internal shifts, and preserve finitely generated graded projectives. $I n d$ is left adjoint to $R e s$ . The right adjoint of $R e s$ is $C o i n d = {HOM}_{B} (e A, -)$ , with $(a \cdot f) (t) = f (t a)$ , and the exact generic second-adjunction formula is ${C o i n d}_{β, γ} (M ⊠ N) ≅ {I n d}_{γ, β} (N ⊠ M) {c} .$ Thus the second adjunction reverses the two blocks and has the displayed shift; same-ordered unshifted biadjunction is not asserted. The formula is an isomorphism of functors and holds on arbitrary graded $B$ -modules after interpreting the right side with the block-flip equivalence. This is distinct from the cyclotomic $E_{i} / F_{i}$ biadjunction theorem. Moreover $e R (β + γ) e^{'}$ has a finite Mackey filtration indexed by nonnegative four-block weight matrices with row sums $β, γ$ and column sums $β^{'}, γ^{'}$ , whose subquotient is the corresponding composite four-block restriction, flip of the middle blocks and induction bimodule, shifted by $- (b ∣ c)$ ; the same statement holds for arbitrary numbers of blocks.

Facts & Assumptions

Given: A KLR datum $Q$ , weights $β, γ \in Q_{+}$ , the algebra $A = R (β + γ)$ with idempotent $e = e (β, γ)$ , the tensor algebra $B = R (β) \otimes_{k} R (γ)$ , the bimodules $e A$ and $A e$ , and the functors $I n d = A e \otimes_{B} -$ , $R e s = e (-)$ of Induction and restriction bimodules for KLR algebras.

[F1]

PBW basis (The PBW (monomial) basis theorem for KLR algebras): monomials $τ_{w} x_{1}^{a_{1}} \dots x_{n}^{a_{n}} e (ν)$ form a basis; every permutation $w$ has a unique decomposition $w = w_{P} w_{shuffle}$ as a product of a parabolic element of $S_{∣ β ∣} \times S_{∣ γ ∣}$ and a minimal-length shuffle, with additive length.

[F2]

Exactness and projectives (Bimodule tensor exactness and preservation of finite projectives have separate hypotheses): if $M$ is flat as an underlying right $A$ -module then $M \otimes_{A} -$ is exact on graded left modules; if $M$ is finite graded projective as a left $B$ -module then $M \otimes_{A} -$ carries finite graded projective left $A$ -modules to finite graded projective left $B$ -modules.

[F3]

Tensor-Hom adjunction (Associative and graded bimodule tensor–Hom adjunction): ${Hom}_{B, 0} (M \otimes_{A} X, Y) ≅ {Hom}_{A, 0} (X, {HOM}_{B} (M, Y))$ naturally, with $HOM$ the graded hom of finite sums of homogeneous maps.

[F4]

Shifts (Graded balanced tensor product and homogeneous Hom): the balanced tensor product carries the total-degree internal grading, and outer actions by graded algebras are homogeneous; all shifts below are internal shifts.

Proof

technique · direct; decompose $Ae$ as a right block module and $eA$ as a left block module, then establish exactness, projectivity, adjunctions and the double-coset filtration

1.1givenF1algebra

(Shuffle decomposition and freeness.) Choose reduced PBW words adapted to the parabolic subgroup $S_{∣ β ∣} \times S_{∣ γ ∣}$ . For each concatenated word $ξ = ν * η$ and minimal right coset representative $w$ , the vectors $τ_{w} e (ξ) b$ , with $b$ ranging through the PBW basis of $e (ξ) B$ , form precisely the PBW basis of the right $B$ -module $A e$ : the permutation is a minimal shuffle followed by an internal block permutation, with additive length, and the right idempotent records the source word in the corner $e$ . Thus $A e ≅ ⨁_{w, ξ} e (ξ) B {δ_{R} (w, ξ)}$ , with the displayed degree of $τ_{w} e (ξ)$ . Reversal of products fixing dots, crossings and idempotents preserves the defining relations and is an anti-involution; it sends $A e$ to $e A$ and $τ_{w} e (ξ)$ to $e (ξ) τ_{w^{- 1}}$ , giving the left $B$ -module decomposition of $e A$ into the corresponding $B e (ξ)$ with shifts $δ_{L} (w, ξ) = \deg (e (ξ) τ_{w})$ . Both decompositions are finite sums of principal graded projectives. Ungraded, for each shuffle the sum over all concatenated $ξ$ is one copy of $B$ , so both ranks are $(\binom{∣ β ∣ + ∣ γ ∣}{∣ β ∣})$ . Neither bimodule is replaced by $e A e$ : its unconstrained opposite endpoint is essential for the shuffle basis.

1.2givenF3algebra

(Left adjunction.) For a graded left $B$ -module $X$ and a graded left $A$ -module $L$ , currying gives the natural degree-zero bijection ${Hom}_{A, 0} (A e \otimes_{B} X, L) \to {Hom}_{B, 0} (X, e L)$ , $Φ \mapsto (x \mapsto Φ (e \otimes x))$ , with inverse $f \mapsto (a e \otimes x \mapsto a f (x))$ , well defined on balanced tensors and inverse by the idempotence of $e$ ; this is the adjunction of [F3] for the $(A, B)$ -bimodule $A e$ .

2.1step 1.1F2F4algebra

(Exactness and projectivity.) The right $B$ -module $A e$ is projective, hence flat, by step 1.1, so $I n d = A e \otimes_{B} -$ is exact by [F2]. Idempotent restriction is exact on left modules: for a surjection $L \to N$ , a vector $e n \in e N$ lifts to $e ℓ \in e L$ , and kernels and images commute with multiplication by the degree-zero idempotent $e$ ; no right action on $L$ is required. A finitely generated graded projective left $B$ -module is a summand of a finite sum of shifts of $B$ , so its induction is a summand of the corresponding sum of shifts of $A e$ . As a left $A$ -module $A e$ is the image of the degree-zero idempotent endomorphism $a \mapsto a e$ of $A$ , hence is finitely generated graded projective. Similarly, if $P$ is a finitely generated graded projective left $A$ -module, realize $P$ as a summand of a finite sum of shifts of $A$ and apply $e (-)$ to its split inclusion and projection. Then $e P$ is a summand of the corresponding sum of shifts of $e A$ , which is finitely generated graded projective as a left $B$ -module by step 1.1. Internal shifts commute with induction and restriction because their balance relations and idempotent action have degree zero. This proves exactness, shift compatibility and both finite-projective preservation clauses.

2.2step 1.1F1F3algebra

(Coinduction and the longest shuffle.) The right adjoint of $R e s$ is $C o i n d = {HOM}_{B} (e A, -)$ by [F3] applied to $M = e A$ , and explicit currying gives $(a \cdot f) (t) = f (t a)$ . For the second-adjunction formula let $y$ be the longest block shuffle of the two blocks, of degree $- c$ (each of the $∣ β ∣ ∣ γ ∣$ crossed positions contributes the negative pairing of the two colours), and let $e^{'} = e (γ, β)$ be the reversed-block idempotent. By PBW there is a well-defined $B$ -balanced coefficient map $t : e A e^{'} \to B$ projecting onto the $y$ -coefficient, where the reversed block order is read in $B$ through the flip equivalence $R (β) \otimes_{k} R (γ) \to R (γ) \otimes_{k} R (β)$ ; dot slides and block crossings preserve the longest-shuffle coefficient, while every dot or braid correction in a shorter shuffle has strictly smaller shuffle length and is killed by $t$ . Hence the map $Φ : A e^{'} \to {HOM}_{B} (e A, B)$ , $z \mapsto (u \mapsto t (u z))$ , is a degree- $c$ map of $(A, B)$ -bimodules, and it is compatible with the flip identification.

2.3step 1.1F1algebra

(Mackey filtration.) Fix a second flag with idempotent $e^{'} = e (β^{'}, γ^{'})$ and choose reduced double-coset representatives for the row block permutation group and the column block permutation group acting on the $∣ β ∣ + ∣ γ ∣$ positions. Partition the PBW monomials of the corner $e A e^{'}$ by the four intersection blocks $(a, b; c, d)$ with row sums $β, γ$ and column sums $β^{'}, γ^{'}$ , and filter by the crossing length of the double-coset representatives. The span of the double cosets of strictly smaller length is a $(B, B^{'})$ -sub-bimodule: sliding a block generator across a representative either stays in its double coset with an internal block permutation, or produces a quadratic/braid correction of strictly smaller length, which lies in the smaller span. In the associated quotient only the monomials whose permutation is the minimal double-coset representative survive; they are exactly tensor products of the PBW bases of the four blocks composed with the row and column induced bimodules and the unique minimal crossing of the two middle blocks, which carries the shift $- (b ∣ c)$ because the crossed colours are precisely $b$ and $c$ . The basis bijection defines inverse maps between the associated quotient and the composite four-block restriction/induction bimodule, so the filtration is proved and not merely a character identity; iterating the same permutation argument gives the multi-block statement.

3.1step 2.2F1F3algebra

(Triangularity and the functor formula.) Pair the right $B$ -module shuffle basis of $A e^{'}$ (through the block flip) with the left $B$ -module shuffle basis of $e A$ by complementary shuffles. Ordered by shuffle length, the coefficient matrix of the pairing $u \mapsto t (u z)$ is triangular: a complementary pair is exactly the reduced longest shuffle with coefficient $1$ up to the absorbed equal-colour signs, and a noncomplementary pair either misses its permutation or reduces by a quadratic or braid correction to strictly smaller length, hence vanishes in the associated graded. Finite triangular elimination over $B$ (division by the diagonal units $1$ ) inverts the matrix, so $Φ$ is an isomorphism of graded $(A, B)$ -bimodules; consequently ${HOM}_{B} (e A, B) ≅ A e^{'} {c}$ with the $B$ -action transported by the block flip. Since $e A$ is finite projective over $B$ by step 1.1, evaluation identifies ${HOM}_{B} (e A, B) \otimes_{B} X ≅ {HOM}_{B} (e A, X)$ for every graded $B$ -module $X$ (checked on finite free modules and finite summands), and tracking the outer right $B$ -action through the flip gives the functorial isomorphism ${C o i n d}_{β, γ} (M ⊠ N) ≅ {I n d}_{γ, β} (N ⊠ M) {c}$ .

4.1step 1.1step 2.1step 1.2step 2.2step 3.1step 2.3algebra∎

All clauses of the statement are now proved: step 1.1 gives the freeness and the labelled shuffle shifts, step 2.1 the exactness, shift compatibility and projective preservation, step 1.2 the left adjunction, steps 2.2 and 3.1 the right adjoint and its exact reversed-block formula, and step 2.3 the finite Mackey filtration and its multi-block version.

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Nil-Hecke idempotents and divided-power KLR projectives

Definition

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum. For $i \in I$ and $n \geq 1$ the subalgebra $R (n α_{i})$ is generated by the polynomial subalgebra $k [x_{1}, \dots, x_{n}]$ and the crossings $τ_{1}, \dots, τ_{n - 1}$ with $τ_{a}^{2} = 0$ (because $Q_{i i} = 0$ ), the braid and far commutation relations, and the nil-Hecke dot moves, i.e. it is the nil Hecke algebra of rank $n$ ; its PBW basis is ${x_{1}^{a_{1}} \dots x_{n}^{a_{n}} τ_{w}}_{a \in Z_{\geq 0}^{n}, w \in S_{n}}$ (The PBW (monomial) basis theorem for KLR algebras). Fix a reduced expression $w_{0} = s_{i_{1}} \dots s_{i_{(\binom{n}{2})}}$ of the longest element and put $\partial_{w_{0}} : = τ_{i_{1}} \dots τ_{i_{(\binom{n}{2})}}$ (independent of the reduced expression). The divided-power idempotent is $e_{i, n} : = x_{1}^{n - 1} x_{2}^{n - 2} \dots x_{n - 1} \partial_{w_{0}} \in R (n α_{i}),$ and its horizontal reflection is $ψ (e_{i, n}) : = \partial_{w_{0}} x_{1}^{n - 1} \dots x_{n - 1}$ . Both are nonzero idempotents of degree $0$ , and $e_{i, n} \neq 0$ because the PBW basis separates the monomials involved. The divided-power projective is $P_{i, n} : = R (n α_{i}) ψ (e_{i, n}) {- d_{i} (\binom{n}{2})},$ where the shift is the one fixed by the source normalization: as a graded left module $R (n α_{i}) ψ (e_{i, n})$ is the polynomial representation, it is indecomposable projective, and with this shift $R (n α_{i})$ is a finite direct sum of $[n]_{i}!$ shifted copies of $P_{i, n}$ , where $[n]_{i}! = \prod_{a = 1}^{n} \frac{q_{i}^{a} - q_{i}^{- a}}{q_{i} - q_{i}^{- 1}}$ is the $q_{i}$ -factorial of Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ (at $q_{i} = 1$ it is $n!$ ); hence $P_{i, n}$ is the unique indecomposable projective summand up to shift and ${E n d}_{R (n α_{i})} (P_{i, n})_{0} = k$ . For a divided-power sequence $θ = i_{1}^{(n_{1})} \dots i_{r}^{(n_{r})}$ put $β = \sum_{a} n_{a} α_{i_{a}}$ and $P_{θ} : = I n d (P_{i_{1}, n_{1}} ⊠ \dots ⊠ P_{i_{r}, n_{r}})$ along the ordered concatenation of the blocks of Induction and restriction bimodules for KLR algebras; then $[P_{θ}]$ represents the ordered product of these divided powers; one may reorder only by the proved commuting/Serre relations, adjacent colours need not commute, and for a single colour $θ = i^{(n)}$ one has the indecomposability above.

Projective duality. For a finite graded projective left $R (β)$ -module $P$ put $P^{#} : = {HOM}_{R (β)} (P, R (β))^{ψ}$ , the $ψ$ -twisted graded Hom into the rank-one free module, where $ψ$ is the horizontal-reflection anti-involution. Evaluation $P^{#} \to$ (dual of $P$ ) identifies $(P^{#})^{#} ≅ P$ for finite shifted free modules, and hence for every finite graded projective; the duality reverses internal shifts and fixes each normalized $P_{i, n}$ after the displayed shift.

Remark

Nil-Hecke matrix calculation. Put $V = k [x_{1}, \dots, x_{n}]$ , $C = k [e_{1}, \dots, e_{n}] = V^{S_{n}}$ and $N = (\binom{n}{2})$ . The elementary symmetric functions are algebraically independent and generate the symmetric ring: subtracting the leading symmetric monomial orbit successively expresses every invariant as a polynomial in them, and their distinct leading monomials give independence. Adjoin $x_{1}$ as a root of the monic polynomial $T^{n} - e_{1} T^{n - 1} + \dots + (- 1)^{n} e_{n}$ , then $x_{2}$ as a root of its quotient by $T - x_{1}$ , and continue. Eliminating coefficients recovers the polynomial ring in the roots; each monic extension is free on powers below its degree. Hence $V$ is free over $C$ on $x^{a}$ with $0 \leq a_{r} \leq n - r$ .

Write $δ = (n - 1, n - 2, \dots, 0)$ . Expanding the divided differences gives $\partial_{w_{0}} (f) = \sum_{w \in S_{n}} sgn (w) w (f) / \prod_{a < b} (x_{a} - x_{b})$ : the braid relations make this independent of the reduced word, and the numerator of $x^{δ}$ is the Vandermonde determinant. Thus $\partial_{w_{0}} (x^{δ}) = 1$ , its values are symmetric, and it kills every other Artin basis monomial by degree. It is therefore the coefficient-of- $x^{δ}$ trace of the monic tower. This trace has a perfect $C$ -bilinear pairing $⟨ f, g ⟩ = \partial_{w_{0}} (f g)$ : for a monic extension of degree $m$ , the coefficient-of- $T^{m - 1}$ trace has matrix zero below the antidiagonal and ones on it in the power basis, hence determinant $\pm 1$ ; composing these perfect trace pairings through the tower preserves perfection. Choose the finite trace-dual Artin basis $(b_{a}^{*})$ to $(b_{a})$ . The operators $f \mapsto b_{a} \partial_{w_{0}} (b_{b}^{*} f)$ are matrix units, all lying in $N H_{n}$ ; faithfulness of Faithful polynomial representation of a KLR algebra therefore identifies $N H_{n} = {End}_{C} (V)$ in every characteristic.

Idempotents, column and duality. In this faithful action, $E = \partial_{w_{0}} x^{δ}$ sends $f$ to $\partial_{w_{0}} (x^{δ} f) \in C$ and fixes $C$ , so $E^{2} = E$ . Reflection $ψ$ fixes dots and crossings and reverses products, giving $e_{i, n} = ψ (E)$ also idempotent and nonzero. Both have degree zero. The map $N H_{n} E \to V$ , $a E \mapsto a (1)$ , is an isomorphism with inverse $f \mapsto m_{f} E$ : $E$ is the rank-one projection onto $C \cdot 1$ , so $a E = m_{a (1)} E$ . Its endomorphism ring is $C$ and its degree-zero part is $k$ , proving graded indecomposability. The trace pairing has degree $- 2 d_{i} N$ and satisfies $⟨ a f, g ⟩ = ⟨ f, ψ (a) g ⟩$ : this is immediate for dots and follows for $\partial_{r}$ by expanding $(1 - s_{r}) / (x_{r} - x_{r + 1})$ in the trace formula. Consequently $V^{#} ≅ V {- 2 d_{i} N}$ , and $P_{i, n} = V {- d_{i} N}$ is self-dual. The Artin basis degrees yield $N H_{n} ≅ ⨁_{a} V {- 2 d_{i} ∣ a ∣}$ , since ${End}_{C} (V) = V \otimes_{C} V^{*}$ ; centering gives multiplicity $q_{i}^{N} \prod_{m = 1}^{n} (1 + q_{i}^{- 2} + \dots + q_{i}^{- 2 (m - 1)}) = [n]_{i}!$ . This also proves uniqueness of the indecomposable projective shift orbit: a finite graded projective over the connected positively graded ring $C$ is shifted-free, by lifting a basis modulo $C^{+}$ and using its projective splitting and the least-degree argument on kernel and cokernel, and the matrix equivalence transports that assertion to $N H_{n}$ .

Duality and induction. For a finite graded projective $B = R (β) \otimes R (γ)$ -module $M$ , finite-projective evaluation gives ${HOM}_{R} (R e \otimes_{B} M, R) ≅ {HOM}_{B} (M, e R) ≅ {HOM}_{B} (M, B) \otimes_{B} e R$ . Twisting by reflection yields $(Ind M)^{#} ≅ Ind (M^{#})$ with the same block order and grading. The identities are checked on finite shifted-free $M$ and then its summands. Thus ordered inductions of the normalized one-color columns are self-dual, as required for bar compatibility.

Two colours and order. For several colours $P_{θ}$ is defined by induction along the ordered blocks, so the class is the ordered induction product; blocks of orthogonal colors commute by the invertible degree-zero block shuffle, since their $Q_{i j}$ are nonzero constants, while adjacent colours are related only by the Serre relations proved later on this page, so no silent symmetrization of the word $θ$ is made. The construction selects no representatives beyond the fixed reduced-expression system of The PBW (monomial) basis theorem for KLR algebras and uses no choice principle.

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Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules

Statement

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum and $β \in Q_{+}$ . (i) Every finitely generated graded left $R (β)$ -module has a decomposition into finitely many indecomposable graded modules, unique as a finite multiset of degree-zero graded isomorphism classes, including the actual shifts of their summands; in particular the category ${G r M o d}_{0} R (β)$ of finitely generated graded modules has the Krull-Schmidt property. (ii) Every simple graded $R (β)$ -module is finite-dimensional, killed by the augmentation ideal ${S y m}^{+} (β)$ of the CENTRAL color-symmetric polynomial subalgebra $C_{β}$ ; up to isomorphism and shift there are finitely many, and each has a projective cover which is finitely generated. (iii) The split Grothendieck group $K_{0} (R (β))$ of finite graded projective modules is a free $Z [q, q^{- 1}]$ -module: the classes of the indecomposable finitely generated graded projectives, modulo grading shifts, form a basis with the shift action $q [P] = [P {1}]$ ; equivalently the classes of all finitely generated indecomposable graded projectives form a shift-orbit basis. Similarly $G_{0} (R (β))$ is free on the shift orbits of simple modules, and the Hom pairing of projective covers and simples has diagonal entries $\dim_{k} {End}^{0} (S_{b})$ , with the selected cover/simple shifts, under the Hom convention of Projective–module Hom pairing on class generators.

Facts & Assumptions

Given: A KLR datum $Q$ , a weight $β \in Q_{+}$ , the locally finite graded algebra $R (β)$ with its finite-dimensional homogeneous components of The PBW (monomial) basis theorem for KLR algebras, and the shift conventions $(M {r})_{d} = M_{d - r}$ of Associative graded algebras, bimodules, and internal shifts.

[F1]

PBW basis (The PBW (monomial) basis theorem for KLR algebras): the monomials $τ_{w} x_{1}^{a_{1}} \dots x_{n}^{a_{n}} e (ν)$ form a $k$ -basis of $R_{n} (Q)$ , each $R (β)$ has finite-dimensional homogeneous components and is bounded below, and its graded ranks do not depend on the parameters.

[F2]

Finite graded projectives (Finite graded projective modules): finite graded projectivity means projectivity in ${G r M o d}_{0}$ tested against degree-zero epimorphisms, plus finite generation; the zero module is allowed.

[F3]

The finite-dimensional shift-orbit basis theorem (Shift-orbit bases for graded simple and projective classes) applies to a finite-dimensional graded algebra; $R (β)$ is infinite-dimensional (for $β \neq 0$ ), so it is not applied to $R (β)$ itself but only to its finite-dimensional central quotients below.

Proof

technique · direct; establish freeness over the central color-symmetric subalgebra, then Krull-Schmidt and projective covers by finite-dimensional endomorphism-algebra arguments, then the orbit bases

1.1givenF1algebra

(Central polynomial subalgebra and freeness.) Let $C_{β}$ be the color-symmetric polynomial subalgebra: on the summand $e (ν) R (β) e (ν)$ it is the polynomial ring in the elementary symmetric functions of the dots of each color separately. It is central: unequal crossings permute dot positions, and equal-color divided differences $\partial_{a}$ commute with symmetric polynomials in $x_{a}, x_{a + 1}$ by the Leibniz rule $\partial_{a} (f g) = \partial_{a} (f) g + s_{a} (f) \partial_{a} (g)$ applied to $s_{a}$ -invariant $f$ . For a color occurring $m$ times, start with $A_{0} = k [E_{1}, \dots, E_{m}]$ and $f_{0} (T) = T^{m} - E_{1} T^{m - 1} + \dots + (- 1)^{m} E_{m}$ . Successively put $A_{r} = A_{r - 1} [X_{r}] / (f_{r - 1} (X_{r}))$ and $f_{r} (T) = f_{r - 1} (T) / (T - X_{r})$ for $1 \leq r \leq m$ ; the division has zero remainder in $A_{r} [T]$ and $f_{r}$ is monic of degree $m - r$ . Monic division makes $A_{r}$ free of rank $m - r + 1$ over $A_{r - 1}$ , with basis $1, X_{r}, \dots, X_{r}^{m - r}$ . In $A_{m}$ the identity $f_{0} (T) = \prod_{r = 1}^{m} (T - X_{r})$ identifies $E_{j}$ with the elementary symmetric polynomial in the $X_{r}$ . Thus $k [x_{1}, \dots, x_{m}] \to A_{m}$ , $x_{r} \mapsto X_{r}$ , is surjective, and its inverse sends $X_{r} \mapsto x_{r}$ and $E_{j}$ to that elementary symmetric polynomial: each successive root relation holds under this substitution. Consequently $k [x_{1}, \dots, x_{m}]$ is free of rank $m!$ over its symmetric coefficient subring, on the Artin monomials $x_{1}^{a_{1}} \dots x_{m}^{a_{m}}$ with $0 \leq a_{r} \leq m - r$ ; only the first extension, not the full polynomial ring, has rank $m$ . Combining with the PBW basis [F1], whose dot monomials factor over the color blocks, gives that $R (β)$ is a finite free $C_{β}$ -module, of rank $(∣ β ∣!)^{2}$ per weight counting the permutation basis; only centrality and this freeness are used below.

2.1step 1.1F2algebra

(Krull-Schmidt and splitting.) Every finitely generated graded module is bounded below and locally finite by [F1]. Choose finitely many homogeneous generators of degrees $d_{j}$ ; evaluation on the generators injects ${End}^{0} (M)$ into the finite product of the finitely many finite-dimensional pieces $M_{d_{j}}$ , so every degree-zero endomorphism algebra is finite-dimensional, and a nontrivial homogeneous idempotent splits $M$ into two nonzero summands with strictly smaller endomorphism algebras; splitting terminates. For an indecomposable $M$ , every endomorphism $f$ satisfies a polynomial minimal equation because ${End}^{0} (M)$ is finite-dimensional; factoring it as $T^{r} g (T)$ with $g (0) \neq 0$ , Bezout gives a polynomial projector separating the nilpotent part from the invertible part, so indecomposability forces $f$ or $1 - f$ to be invertible. Hence the nonunits form the unique maximal ideal, and the usual comparison of two decompositions $⨁ p_{i} i_{j} p_{j} = i d$ produces a matching summand and cancellation, proving uniqueness in the degree-zero category; the shifts are recorded as part of each summand.

3.1step 1.1step 2.1F1algebra

(Simples and covers.) For a graded simple $S$ and a positive-degree central $c \in C_{β}^{+}$ , the kernel and image of $c$ are graded submodules; if $c$ acted nontrivially it would be injective and hence surjective, contradicting the least nonzero degree of the bounded-below module $S$ ; so $C_{β}^{+} S = 0$ and $S$ is finite-dimensional because $\overset{ˉ}{R} = R (β) / C_{β}^{+} R (β)$ is finite-dimensional by step 1.1. The finite-dimensional basis theorem [F3] applies to $\overset{ˉ}{R}$ : it has finitely many graded simple shift orbits, and the same then holds for $R (β)$ , since any simple $R (β)$ -module is one. Projective covers are then lifted to $R (β)$ : choose a finite shifted-free $R$ -module $F$ whose reduction $\overset{ˉ}{F}$ surjects onto the projective cover $\overset{ˉ}{Q} ↠ \overset{ˉ}{S}$ . That surjection splits by projectivity of $\overset{ˉ}{Q}$ , giving an idempotent of ${End}_{\overset{ˉ}{R}}^{0} (\overset{ˉ}{F})$ ; entrywise reduction ${End}_{R}^{0} (F) \to {End}_{\overset{ˉ}{R}}^{0} (\overset{ˉ}{F})$ is surjective; the reduction kernel in ${End}_{R}^{0} (F)$ is nilpotent because its entries have positive central degree and a long product has degree exceeding every fixed difference of generator degrees minus the lower bound of $F$ , so the projector onto $\overset{ˉ}{Q}$ is lifted by the finite nilpotent-ideal idempotent-lifting iteration $e \mapsto e - (2 e - 1) (e^{2} - e)$ ; the image $Q$ of the lifted projector is projective, its kernel is graded superfluous (after reduction by $C^{+}$ it is superfluous for $\overset{ˉ}{Q}$ , and a bounded-below graded quotient $X = C^{+} X$ vanishes by its least nonzero degree), so $Q$ is an indecomposable projective cover, and every indecomposable projective is such a cover up to shift.

4.1step 2.1step 3.1F3algebra∎

(Orbit bases.) Finite decompositions into indecomposables give that $K_{0} (R (β))$ is free on the shift orbits of indecomposable finite graded projectives, with $q [P] = [P {1}]$ , and finite length of finite-dimensional modules gives that $G_{0} (R (β))$ is free on the simple shift orbits; both are $Z [q, q^{- 1}]$ -modules in the sense of Graded Grothendieck groups, shift action, and Cartan map, the shift action is free because a nonzero finite-dimensional graded module is not isomorphic to a nonzero shift of itself (compare its least and greatest degrees), and the finite-dimensional theorem [F3] gives the same statement for the quotient categories. For a projective cover $P_{b}$ and simple $S_{c}$ with their selected shifts, $Hom (P_{b}, S_{c}) = 0$ unless $b = c$ at the selected shift, when it equals $End (S_{b})$ by the uniqueness of the cover, whose degree-zero endomorphism ring is a division $k$ -algebra by Schur's argument; finite-dimensionality alone does not identify it with $k$ ; this is the diagonal-entry computation of the Hom pairing of Projective–module Hom pairing on class generators. The stronger statement that the diagonal scalar rings equal $k$ over every field, and the associated unscaled dual-basis conclusion, are not inferred here from finite-dimensional algebra alone and are proved separately later on this page; nothing above uses them.

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Maximal KLR tails give a unique induction head and absolute simple modules

Statement

Let $R$ be any graded KLR datum in this page over any field $k$ . For a finite-dimensional graded module $M$ of weight $β$ set $ϵ_{i} (M) = \max {r : e (β - r α_{i}, r α_{i}) M \neq 0}$ . If $S$ is simple and $r = ϵ_{i} (S) > 0$ , there is a simple graded $R (β - r α_{i})$ -module $N$ with $ϵ_{i} (N) = 0$ such that $Δ_{i}^{r} S = N ⊠ L (i^{r})$ , where $L (i^{r})$ is the unique nil-Hecke simple up to shift. Conversely, for any such $N$ , $I n d (N ⊠ L (i^{r}))$ has a unique simple graded head $S$ , appearing once as a composition factor, with $ϵ_{i} (S) = r$ and $Δ_{i}^{r} S = N ⊠ L (i^{r})$ ; every other composition factor has smaller $i$ -tail. The graded simples of every $R (β)$ remain simple without grading and are absolutely simple; all ordinary endomorphisms are $k$ and nonzero homogeneous endomorphisms have degree $0$ . The classes of normalized simples and their projective covers are therefore unscaled dual shift-orbit bases for the projective/simple Hom pairing. For each $β$ there is a finite family of ordered divided-power words $Y_{b}$ whose normalized projective classes pair triangularly with the simple classes, with diagonal entries $1$ after choosing the simple grading shifts. Thus the selected $[P (Y_{b})]$ form an $A$ -basis of $K_{0} (R (β))$ ; arbitrary divided words span, but need not be a basis. The identical statements hold for maximal FIRST-color blocks, with $ϵ_{i}^{L} (M) = \max {r : e (r α_{i}, β - r α_{i}) M \neq 0}$ , restriction $Δ_{i}^{(L, r)}$ and induction $I n d (L (i^{r}) ⊠ N)$ .

Facts & Assumptions

Given: A graded KLR datum over any field $k$ , the functors $Δ_{i} = {R e s}_{β - r α_{i}, r α_{i}}$ and $I n d$ of Generic KLR induction and restriction adjunctions preserve projectives, the finite-dimensional graded simples of Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules, and the nil-Hecke idempotents and simple $L (i^{r})$ of Nil-Hecke idempotents and divided-power KLR projectives.

[F1]

With $C_{r} = k [x_{1}, \dots, x_{r}]^{S_{r}}$ , the full nil-Hecke algebra $N H_{r} = R (r α_{i})$ is the graded endomorphism algebra of the Artin polynomial column, a free $C_{r}$ -module of rank $r!$ ; thus it is a graded matrix algebra over $C_{r}$ , with the corresponding row and column shifts. Its positive-center augmentation quotient is a graded matrix algebra over $k$ , with its unique graded simple $L (i^{r})$ absolutely simple up to shift. Every finite-dimensional graded simple over $R (μ) \otimes_{k} N H_{r}$ is killed by that positive-center ideal and is, up to shift, uniquely of the form $N ⊠ L (i^{r})$ for a graded simple $R (μ)$ -module; the center-annihilation argument is supplied in step 1.1.

[F2]

Exactness, the adjunctions and the Mackey filtration of Generic KLR induction and restriction adjunctions preserve projectives: $I n d$ is left adjoint to $R e s$ , both are exact, and a corner restriction has a finite filtration whose subquotients are composites of four-block restrictions, flips and inductions with the displayed shifts.

[F3]

Krull-Schmidt and covers (Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules): finite-dimensional graded modules have finite unique decompositions, simples are finite-dimensional, and the projective/simple Hom pairing has diagonal entries $\dim_{k} {End}^{0} (S_{b})$ .

Proof

technique · direct; identify the nil-Hecke tail by matrix units, compute the maximal restriction through the Mackey filtration, then run the triangular basis argument

1.1givenF1algebra

(Nil-Hecke tail and its matrix units.) The constant-colour block $N H_{r} = R (r α_{i})$ acts on its polynomial column over $C_{r} = k [x_{1}, \dots, x_{r}]^{S_{r}}$ . The Artin monomial basis and perfect symmetric-variable-linear trace of Nil-Hecke idempotents and divided-power KLR projectives give integral matrix units and identify the full algebra with graded endomorphisms of that free rank- $r!$ column, hence with matrices over $C_{r}$ carrying the Artin degree shifts. Reducing the positive-degree central elementary symmetric functions to zero gives the graded matrix quotient over $k$ ; its column simple $L (i^{r})$ is absolutely simple, and the divided idempotent cuts out a one-dimensional image. Its full-algebra polynomial-column graded projective cover has degree-zero endomorphism ring $k$ , although its entire endomorphism ring is $C_{r}$ . If $T$ is a finite-dimensional graded simple over $R (μ) \otimes_{k} N H_{r}$ , a positive-degree central operator has graded kernel and image. A nonzero such operator would therefore be invertible by graded simplicity, which is impossible on the finite nonzero degree support of $T$ . Thus the positive tail center annihilates $T$ , and it factors through $R (μ) \otimes_{k} {Mat}_{r!} (k)$ with the indicated shifts. Matrix units identify $T$ with $N ⊠ L (i^{r})$ for a graded simple $R (μ)$ -module $N$ , unique up to shift. This uses the augmentation matrix quotient for finite simples, not a false identification of the full nil-Hecke algebra with matrices over the ground field.

2.1step 1.1F2algebra

(Maximal restriction.) Let $S$ be a finite-dimensional graded simple with $r = ϵ_{i} (S) > 0$ maximal. The finite-dimensional module $Δ_{i}^{r} S$ is nonzero by the definition of $ϵ_{i}$ and has a simple submodule; by step 1.1 it is of the form $N ⊠ L (i^{r})$ for a graded simple $R (β - r α_{i})$ -module $N$ . If $N$ carried a word ending in $i$ , then concatenating any nonzero nil-Hecke tail word would produce a nonzero vector in $S$ of tail $r + 1$ , contradicting maximality, so $ϵ_{i} (N) = 0$ . By the left adjunction of [F2] the inclusion $N ⊠ L (i^{r}) ↪ Δ_{i}^{r} S$ corresponds to a nonzero map $M : = I n d (N ⊠ L (i^{r})) \to S$ , which is surjective since $S$ is simple. In the Mackey filtration of $Δ_{i}^{r} M$ every shuffle bringing $t > 0$ colours $i$ from the first block into the final tail contains a factor $Δ_{i}^{t} N$ and is zero; the only surviving filtration quotient is $N ⊠ L (i^{r})$ with zero crossing shift, so $Δ_{i}^{r} M = N ⊠ L (i^{r})$ exactly. This is a one-surviving-quotient computation, not a semisimplicity assumption.

3.1step 2.1F1F3algebra

(Unique head and simplicity.) Every nonzero quotient $Q$ of $M$ has nonzero defining corner: if $e Q = 0$ with $e = e (β - r α_{i}, r α_{i})$ , then the composite $A e \otimes_{B} (N ⊠ L (i^{r})) \to M \to Q$ is zero by the tensor/Res adjunction, hence $Q = 0$ . The finite graded semisimple head of $M$ therefore has exactly one simple summand, occurring once, because its exact $e$ -restriction is a quotient of the simple $e M = N ⊠ L (i^{r})$ ; this head is $S$ , and $e M \to e S$ is an isomorphism, so exactness gives $e \ker (M \to S) = 0$ and every other composition factor of $M$ has smaller tail. Underlying and absolute simplicity of $S$ : induct on the height of $β$ . For positive height choose $i$ with $r > 0$ ; by step 2.1 and induction $N$ and $L (i^{r})$ are absolutely simple, so $e S = N ⊠ L (i^{r})$ has endomorphism ring $k$ ; if $U \leq S$ is any ungraded submodule with $e U \neq 0$ , then $e U = e S$ and $U = S$ because $S$ is generated by $e S$ ; if $e U = 0$ and $U \neq 0$ , filter $S$ by $F_{d} S = ⨁_{j \leq d} S_{j}$ and $U$ by $F_{d} U = U \cap F_{d} S$ . Every homogeneous algebra element of degree $a$ sends $F_{d} U$ into $F_{d + a} U$ , so $gr U$ embeds as a nonzero graded submodule of $gr S = S$ . Since $e$ has degree $0$ and annihilates $U$ , it annihilates $gr U$ ; graded simplicity gives $gr U = S$ , contradicting $e S \neq 0$ . Thus $S$ is simple as an ungraded module, and every $R$ -endomorphism of $S$ is determined on $e S$ , giving ${End}_{R} (S) = k$ . The finite-density matrix argument then shows absolute simplicity over every field: any $m$ linearly independent vectors can be sent to arbitrary targets by one algebra element, by induction on $m$ and the commutant $k$ clause, so the action realizes all $k$ -linear matrices and remains simple after field extension.

4.1step 3.1F3algebra

(Triangular divided-string basis.) Choose a cyclic order of the finite colour support and recursively strip maximal $i$ -tails, producing for each simple shift orbit a finite tuple $Y_{b}$ of ordered divided powers; two simples with the same tuple agree by the unique-head recursion of step 3.1. Form $P (Y_{b})$ from the reversed ordered normalized divided blocks. Its homogeneous Hom to $S_{c}$ is the corresponding product of divided idempotents acting on $S_{c}$ ; comparing tuples lexicographically, if $Y_{b} > Y_{c}$ at the first differing entry then the required tail exceeds the maximal tail of the successively stripped simple $S_{c}$ , and the corner is zero. When the tuples agree, every maximal restriction is the simple tensor computed in step 2.1 and each nil-Hecke divided idempotent reduces the tail to one dimension, so by induction the diagonal Hom is a single Laurent monomial with coefficient $1$ . Shifting each $S_{b}$ makes the diagonal $1$ . The finite matrix is triangular over $A = Z [q, q^{- 1}]$ with diagonal $1$ ; Krull-Schmidt and the projective-cover/simple Hom pairing (now with $End (S) = k$ from step 3.1) identify it with the matrix expressing the projective classes in the cover basis, and successive integral triangular elimination proves that the selected classes $[P (Y_{b})]$ are an $A$ -basis of $K_{0} (R (β))$ ; the whole family of divided words spans because the recursion covers every simple, while redundancy of non-selected words is allowed. This completes the integral surjectivity argument without a PBW rank or flat-family shortcut.

5.1step 1.1step 3.1step 4.1F2algebra∎

(First-color version.) For maximal FIRST-color blocks use the defining idempotent $e (r α_{i}, μ)$ and $M = I n d (L (i^{r}) ⊠ N)$ with $ϵ_{i}^{L} (N) = 0$ ; a simple submodule of the maximal first-block restriction $Δ_{i}^{(L, r)} S$ has the form $L (i^{r}) ⊠ N$ by the same nil-Hecke matrix units of step 1.1, and any first-colour support of $N$ would extend the maximal first block, so it vanishes. In the Mackey restriction of $Δ_{i}^{(L, r)} M$ every nontrivial shuffle bringing a positive number $t$ of $i$ -colours from $N$ into the first block contains $Δ_{i}^{(L, t)} N$ and is zero, so the sole uncrossed quotient is $L (i^{r}) ⊠ N$ with shift $0$ ; the defining-corner generation and exact-head argument of steps 3.1 and 3.1 are unchanged. Thus the first-block assertions are proved directly, with no unsigned star-crystal preservation assumed.

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The KLR Grothendieck ring satisfies the divided-power quantum Serre relations

Statement

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum and let $K_{0} (R) : = ⨁_{β \in Q_{+}} K_{0} (R (β))$ be the split Grothendieck group of finite graded projective modules with the shift action $q [P] = [P {1}]$ (Split Grothendieck group of an additive category, Graded Grothendieck groups, shift action, and Cartan map, Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules). (i) The induction product $[P] [P^{'}] : = [{I n d}_{β, γ} (P ⊠ P^{'})]$ for $P \in K_{0} (R (β)), P^{'} \in K_{0} (R (γ))$ makes $K_{0} (R)$ a unital associative $Z [q, q^{- 1}]$ -algebra, and $[R e s]$ makes it a coassociative counital $Z [q, q^{- 1}]$ -coalgebra; with the twisted multiplication on $K_{0} (R) \otimes K_{0} (R)$ defined by $(x_{1} \otimes x_{2}) (x_{1}^{'} \otimes x_{2}^{'}) = q^{- (∣ x_{2} ∣ ∣ ∣ x_{1}^{'} ∣)} x_{1} x_{1}^{'} \otimes x_{2} x_{2}^{'}$ on homogeneous elements, $[R e s]$ is an algebra homomorphism (Generic KLR induction and restriction adjunctions preserve projectives). (ii) With the divided-power projectives $P_{i, n}$ of Nil-Hecke idempotents and divided-power KLR projectives, the classes $[P_{i, n}]$ satisfy, for all $i \neq j$ , the divided-power quantum Serre relation $\sum_{a + b = - a_{i j} + 1} (- 1)^{a} [P_{i, a}] [P_{j, 1}] [P_{i, b}] = 0 in K_{0} (R),$ with the convention $[P_{i, 0}] = 1$ , and satisfy $[P_{i, 1}] [P_{j, 1}] = [P_{j, 1}] [P_{i, 1}]$ when $a_{i j} = 0$ ; these are the defining relations of the negative half. (iii) Consequently the assignment $θ_{i} \mapsto [P_{i, 1}]$ extends uniquely to an algebra homomorphism $γ : A f \to K_{0} (R)$ of the divided-power form of the negative half, with $γ (θ_{i}^{(n)}) = [P_{i, n}]$ and $γ (θ_{i_{1}}^{(n_{1})} \dots θ_{i_{r}}^{(n_{r})}) = [I n d (P_{i_{1}, n_{1}} ⊠ \dots ⊠ P_{i_{r}, n_{r}})]$ ; $γ$ is compatible with the coproducts ( $r$ on $A f$ and $[R e s]$ on $K_{0} (R)$ ) and with the bar involutions, and it is $Z [q, q^{- 1}]$ -linear.

Facts & Assumptions

Given: A KLR datum $Q$ , its divided-power projectives $P_{i, n}$ of Nil-Hecke idempotents and divided-power KLR projectives, the induction and restriction functors with their exactness, adjunction and Mackey filtration of Generic KLR induction and restriction adjunctions preserve projectives, and the negative half $A f$ with divided powers $θ_{i}^{(n)}$ of The Lusztig divided-power integral form $U_{A}$ and Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ .

[F1]

Corner filtration (Generic KLR induction and restriction adjunctions preserve projectives): $e_{β, γ} R (β + γ) e_{β^{'}, γ^{'}}$ has a finite Mackey filtration indexed by four-block weight matrices whose subquotient is the composite four-block restriction, middle flip and induction bimodule, shifted by the negative pairing of the crossed blocks; Ind is left adjoint to Res and both are exact.

[F2]

Presentation of the negative half (The Lusztig divided-power integral form $U_{A}$ ): $U_{A}^{-}$ is the $A$ -subalgebra generated by the divided powers $F_{i}^{(r)}$ , with the Serre relations; the $A$ -form is spanned by ordered divided words, not by an ordinary free algebra on the $F_{i}$ .

[F3]

The negative half over $Q (q)$ is exactly the algebra presented by the negative Serre relations (Triangular decomposition of a quantized enveloping algebra, clause (ii)); $U_{A}^{-}$ is its $A$ -subalgebra generated by divided powers. No integral shuffle-freeness claim is taken from the formal $C ⟦ ℏ ⟧$ theorem.

Proof

technique · direct; build the induction algebra and restriction coalgebra, then the contracting homotopy for the Serre relation, then descend $\gamma$

1.1givenF1algebra

(Induction algebra and restriction coalgebra.) The induction product is associative and unital: the balanced identifications of Induction and restriction bimodules for KLR algebras for the two parenthesizations of three blocks are inverse degree-zero isomorphisms, and the unit is $[R (0)] = [k]$ . For the block algebra $B$ , the positive-center quotient is the tensor product of the two finite-dimensional graded quotients from Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules. Their simples are absolutely simple by Maximal KLR tails give a unique induction head and absolute simple modules, so their graded semisimple quotients are products of split graded matrix algebras. Tensoring these matrix blocks gives all simple shift orbits of $B$ ; tensor products of the factor projective covers cover them, by reducing their kernels and using the graded least-degree argument. Uniqueness of covers therefore identifies $K_{0} (B)$ with $K_{0} (R (β)) \otimes_{A} K_{0} (R (γ))$ . Restriction is coassociative with counit the weight-zero component, because iterated idempotent restriction multiplies the corresponding block idempotents, and the two orders of multiplication agree. To prove that $[R e s]$ respects the twisted multiplication, apply the Mackey filtration [F1] to the corner $e_{β, γ} R e_{β^{'}, γ^{'}}$ : PBW partitions its basis by the double cosets of the two block permutation groups, ordered by crossing length, and multiplication across a braid move changes only smaller-length terms; the associated quotient is exactly the four-block restriction/induction bimodule shifted by minus the root pairing of the two crossed middle blocks. Taking classes gives $[R e s] (x y) = \sum q^{- (∣ y_{2} ∣ ∣ ∣ x_{1}^{'} ∣)} x_{1} x_{1}^{'} \otimes x_{2} x_{2}^{'}$ with the twisted product, so $[R e s]$ is an algebra homomorphism; no use of the second adjunction is needed.

2.1step 1.1algebraconstruct

(Explicit Serre contraction.) Fix $i \neq j$ , write $d = - a_{i j}$ , $D = d + 1$ , $N = d + 2$ , and $t = [u^{d}] Q_{i j} \neq 0$ . For $0 \leq r \leq D$ , let $E_{r}$ be the product of the reflected nil-Hecke idempotents on the two $i$ -blocks in $i^{r} j i^{D - r}$ , let $s_{r} = - d_{i} ((\binom{r}{2}) + (\binom{D - r}{2}))$ , and put $C_{r} = R E_{r} {s_{r}}$ . Use right multiplication by $a_{r} = E_{r} τ_{N - 1} \dots τ_{r + 1} E_{r + 1}$ for $C_{r} \to C_{r + 1}$ and by $b_{r} = E_{r + 1} τ_{1} \dots τ_{r + 1} E_{r}$ in the reverse direction. Each string crosses $j$ once and respectively $D - r - 1$ or $r$ equal-color pairs, so its degrees are $d_{i} d - 2 d_{i} (D - r - 1)$ and $d_{i} d - 2 d_{i} r$ . As $s_{r + 1} - s_{r} = d_{i} (D - 2 r - 1)$ , both maps have degree zero between the shifted columns. The nil-Hecke identities $τ_{a} E_{r} = 0$ for equal-color crossings internal to its divided blocks and the braid moves give $a_{r - 1} a_{r} = 0$ . At corner $r$ , put $L_{r} = a_{r} b_{r}$ , $R_{r} = b_{r - 1} a_{r - 1}$ , with absent endpoint terms zero. Moving far crossings cancels the common strings in $L_{r} - R_{r}$ ; the remaining braid correction is the divided difference of $Q_{i j}$ . For a monomial $u^{a} v^{b}$ its correction is the sum of $a$ monomials of combined outer degree $a - 1$ , with the native minus sign. The $r - 1$ left and $D - r - 1$ right equal-color divided differences kill it unless $a \geq d$ ; homogeneity permits $a = d$ only when $b = 0$ , and permits no $a > d$ . For $a = d$ exactly the staircase exponents $r - 1, D - r - 1$ survive, each evaluated by the nil-Hecke trace calculation of Nil-Hecke idempotents and divided-power KLR projectives; transporting the variable order on the left block gives $L_{r} - R_{r} = (- 1)^{r} t E_{r}$ , including endpoints. Thus $h_{r} = (- 1)^{r} t^{- 1} b_{r}$ satisfies $a_{r} h_{r} + h_{r - 1} a_{r - 1} = E_{r}$ on every corner. These degree-zero maps contract the finite complex $(C_{r}, a_{r})$ over every field and every (Q1)–(Q4) parameter.

3.1step 2.1algebra

(Alternating classes and the $a_{i j} = 0$ case.) A bounded complex with $d h + h d = 1$ is split contractible: choosing the successive splittings gives a contraction with $h^{2} = 0$ , and then $d + h$ is an isomorphism from even to odd part; hence the alternating sum of the classes of its terms vanishes, which is exactly $\sum_{a + b = D} (- 1)^{a} [P_{i, a}] [P_{j, 1}] [P_{i, b}] = 0$ , the displayed Serre relation. If $a_{i j} = 0$ then the crossing $τ$ of the two colours is invertible (its square is the nonzero constant $Q_{i j}$ ), and right multiplication by it gives a degree-zero isomorphism between the two corners, so $[P_{i, 1}] [P_{j, 1}] = [P_{i j}] = [P_{j i}] = [P_{j, 1}] [P_{i, 1}]$ .

4.1step 1.1step 3.1F2F3algebra∎

(Descent of $γ$ .) First define $γ$ over $Q (q)$ from the presentation of the negative Serre half: [F3] identifies the Serre-presented rational negative half with the actual negative subalgebra, and the assignments $θ_{i} \mapsto [P_{i, 1}]$ , extended multiplicatively and $q$ -linearly, respect all Serre relations by step 3.1 and the commuting relation, so they define an algebra homomorphism $γ_{Q}$ on the localization. Since the $A$ -form is the $A$ -subalgebra spanned by ordered divided words [F2], and $[P_{i, 1}]^{n} = [n]_{i}! [P_{i, n}]$ by the nil-Hecke decomposition, so every rational divided power maps to $[P_{i, n}]$ . These divided-word images lie in $K_{0} (R)$ , which embeds in its rational scalar extension because its shift-orbit basis is free, the homomorphism restricts to the stated $A$ -linear $γ$ ; the formula $γ (θ_{i_{1}}^{(n_{1})} \dots θ_{i_{r}}^{(n_{r})}) = [I n d (P_{i_{1}, n_{1}} ⊠ \dots ⊠ P_{i_{r}, n_{r}})]$ is multiplicativity of $γ$ together with the definition of $P_{θ}$ . Coproduct compatibility is the Mackey computation of step 1.1, and bar compatibility is the projective $^{#}$ duality of Nil-Hecke idempotents and divided-power KLR projectives, which fixes each normalized $P_{i, n}$ and hence each word class. No source parameter-specialization is asserted to preserve projective ranks.

DefinitionDefinition: Literature-sourcedProof: AI-adaptedOpen item page →

The KLR Cartan (graded Hom) pairing on the projective Grothendieck group

Definition

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum and let $ψ$ be the horizontal-reflection anti-involution of $R (β)$ (reflection of diagrams about a horizontal axis, which reverses the order of crossings); for a finitely generated graded left module $P$ let $P^{ψ}$ be the right module with $p \cdot r : = ψ (r) p$ . For $β \in Q_{+}$ define the Cartan pairing $([P], [Q])_{β} : = {g d i m}_{k} (P^{ψ} \otimes_{R (β)} Q) \in Z [q, q^{- 1}] \cdot (β)_{q}, (β)_{q} : = \prod_{i} \prod_{a = 1}^{β_{i}} \frac{1}{1 - q^{2 d_{i} a}},$ for finitely generated graded projective modules $P, Q$ , where ${g d i m}_{k} M = \sum_{m \in Z} q^{m} \dim_{k} M_{m}$ ; the tensor is over the graded algebra with the total-degree grading (Graded balanced tensor product and homogeneous Hom). Equivalently define the hom pairing $([P], [Q])_{β}^{'} : = {g d i m}_{k} {HOM}_{R (β)} (P, Q)$ (Projective–module Hom pairing on class generators, Projective Hom pairing descends and is graded sesquilinear); then $([P], [Q])_{β}^{'} = ([P^{#}], [Q])_{β}$ for the graded left projective $P^{#} : = {HOM}_{R (β)} (P, R (β))^{ψ}$ . Thus the tensor form is symmetric and bilinear; the Hom form is conjugate-linear in its first variable and linear in its second. For the tensor form $([P], q [Q]) = q ([P], [Q])$ and $(q [P], [Q]) = q ([P], [Q])$ , while the Hom form obeys $h (q^{r} [P], q^{s} [Q]) = q^{s - r} h ([P], [Q])$ . Both are additive, vanish on different-weight components, and descend to $K_{0} (R) = ⨁_{β} K_{0} (R (β))$ : the tensor form is $A$ -bilinear and the Hom form is $A$ -sesquilinear (first variable conjugate-linear), with values in $Q (q)$ , more precisely in the localization adjoining all $(1 - q_{i}^{2 a})^{- 1}$ ; the values lie in $Z [q, q^{- 1}] \cdot (β)_{q}$ because $P^{ψ} \otimes_{R (β)} Q$ is a finite free graded module over the central color-symmetric polynomial subalgebra $C_{β}$ of Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules. Normalization: $([P_{i}], [P_{i}]) = (1 - q^{2 d_{i}})^{- 1} = (1 - q_{i}^{2})^{- 1}$ and $([P_{i}], [P_{j}]) = 0$ for $i \neq j$ , where $P_{i} = R (α_{i})$ .

Remarks

Well-definedness. $P^{ψ} \otimes_{R (β)} Q$ is a graded $k$ -vector space of finite dimension in each degree and bounded below, so its graded dimension is a well-defined element of $Z ((q))$ ; the monic-root tower and PBW calculation in step 1.1 of Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules explicitly prove that $R (β)$ is finite graded free over $C_{β}$ , although freeness is not a separate assertion of that lemma's Statement. Write $V = P^{ψ} \otimes_{R} Q$ . Taking the split inclusions of $P$ and $Q$ into finite shifted-free $R$ -modules and tensoring their splitting maps exhibits $V$ as a direct summand of a finite sum of shifts of $R$ . Thus $V$ is finite graded projective over $C_{β}$ . This upgrades the value to $Z [q, q^{- 1}] \cdot (β)_{q}$ : lifting a homogeneous $k$ -basis of $V / C_{β}^{+} V$ gives a surjection from a finite shifted free $C_{β}$ -module with vanishing cokernel, whose projective splitting has kernel $K$ with $K = C_{β}^{+} K$ , also vanishing by the least-degree argument; hence the Hilbert series is a finite Laurent polynomial times $g d i m C_{β} = (β)_{q}$ . Symmetry of the tensor form is commutativity of the balanced tensor $P^{ψ} \otimes_{R} Q ≅ Q^{ψ} \otimes_{R} P$ under the flip; the comparison form is the evaluation isomorphism ${HOM}_{R} (P, R) \otimes_{R} Q \to {HOM}_{R} (P, Q)$ , $f \otimes v \mapsto (p \mapsto f (p) v)$ , an isomorphism for shifted free $P$ and hence for all finite graded projective summands, twisted by $ψ$ on the first factor.

Shifts and the two normalizations. The total Hom-degree convention is that of Projective Hom pairing descends and is graded sesquilinear, so $h (P {r}, Q {s}) = q^{s - r} h (P, Q)$ ; no scalar conjugation of the tensor value is valid, and the rank-one values $(R (α_{i}), R (α_{i})) = 1 / (1 - q_{i}^{2})$ already show that the Hom form is not the plain scalar extension of the tensor form. The generator values follow from $P_{i} = R (α_{i}) = k [x_{1}]$ , $P_{i}^{ψ} \otimes_{R (α_{i})} P_{i} ≅ k [x_{1}]$ with $\deg x_{1} = 2 d_{i}$ , and $([P_{i}], [P_{j}]) = 0$ for $i \neq j$ because the two weight components are orthogonal. The cited projective/module Hom suppliers concern finite-dimensional algebras. Here the same shift and additivity formulas follow directly from homogeneous Hom and evaluation on finite shifted-free modules and summands; their finite-dimensional hypotheses are not applied to $R (β)$ . The anti-involution $ψ$ is recorded explicitly because it fixes the sign of the shift in the Hom form. No choice principle is used.

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The KLR Cartan pairing is the Lusztig-normalized half form

Statement

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum, let $B_{K}$ be the normalized Lusztig-Kashiwara form on the negative half of The Lusztig-Kashiwara bilinear form on the negative half: the recursion and put $B_{L} : = C_{ν} B_{K}$ on the weight component $ν$ , $C_{ν} = \prod_{i} (1 - q_{i}^{2})^{- ν_{i}}$ , the Lusztig-normalized form of The Lusztig–Kashiwara form, parameter and coproduct dictionary. Then for homogeneous $x, y \in A f_{ν}$ the Cartan pairing of The KLR Cartan (graded Hom) pairing on the projective Grothendieck group satisfies $(γ (x), γ (y))_{K L R} = B_{L} (x, y) = C_{ν} B_{K} (x, y),$ where $γ : A f \to K_{0} (R)$ is the algebra homomorphism of The KLR Grothendieck ring satisfies the divided-power quantum Serre relations. Equivalently, the two bilinear forms are equal under the identification of divided-power classes; in particular the KLR pairing is $Q (q)$ -nondegenerate on each finite weight component $A f_{ν} \otimes_{A} Q (q)$ after extending scalars, and $(γ (x), γ (y))_{K L R}^{'} = B_{L} (\overline{x}, y)$ , where bar acts on the first input. On generators: $([P_{i}], [P_{j}]) = δ_{i j} (1 - q_{i}^{2})^{- 1} = B_{L} (θ_{i}, θ_{j})$ and $([P_{i, n}], [P_{i, n}]) = B_{L} (θ_{i}^{(n)}, θ_{i}^{(n)}) = \prod_{a = 1}^{n} (1 - q_{i}^{2 a})^{- 1}$ .

Facts & Assumptions

Given: A KLR datum $Q$ , the Cartan pairing of The KLR Cartan (graded Hom) pairing on the projective Grothendieck group on $K_{0} (R)$ , the homomorphism $γ$ of The KLR Grothendieck ring satisfies the divided-power quantum Serre relations, the normalized half form $B_{K}$ of The Lusztig-Kashiwara bilinear form on the negative half: the recursion with skew derivations $e_{i}^{'}$ of Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half, and the normalization factor $C_{ν} = \prod_{i} (1 - q_{i}^{2})^{- ν_{i}}$ of The Lusztig–Kashiwara form, parameter and coproduct dictionary.

[F1]

Normalization and adjunction (The KLR Cartan (graded Hom) pairing on the projective Grothendieck group): the tensor form is symmetric and bilinear with $([P_{i}], [P_{i}]) = (1 - q_{i}^{2})^{- 1}$ and $([P_{i}], [P_{j}]) = 0$ for $i \neq j$ , and $I n d$ is left adjoint to $R e s$ .

[F2]

Mackey and the coproduct comparison (The KLR Grothendieck ring satisfies the divided-power quantum Serre relations): $[R e s]$ is an algebra homomorphism for the twisted product, so the pairing satisfies the adjunction identities $(x, y y^{'}) = (r (x), y \otimes y^{'})$ and $(x x^{'}, y) = (x \otimes x^{'}, r (y))$ with $r = [R e s]$ .

[F3]

Determination of $B_{K}$ (The Lusztig-Kashiwara bilinear form on the negative half: the recursion): axioms (H), (N), (R): homogeneity, $B (1, 1) = 1$ and $B (F_{i} P, Q) = B (P, e_{i}^{'} Q)$ determine $B$ uniquely, and peeling first letters reduces every evaluation to the base case.

[F4]

Normalization dictionary (The Lusztig–Kashiwara form, parameter and coproduct dictionary): with the source-to-library parameter transport $φ : Q (v) \to Q (q)$ , $φ (v) = q^{- 1}$ the root-colour factor $C_{ν}$ converts the Kashiwara recursion into $B_{L} (F_{i} P, Q) = (1 - q_{i}^{2})^{- 1} B_{L} (P, e_{i}^{'} Q)$ and $B_{L}$ has the generator value $(1 - q_{i}^{2})^{- 1}$ .

[F5]

Nondegeneracy (The negative-half bilinear form is symmetric, nondegenerate and integral, clause (iii)): for every $0 \neq x \in U_{q}^{-}$ there is $y$ with $(x, y) \neq 0$ ; equivalently the form restricted to each finite-dimensional graded piece is nondegenerate. Its construction uses the Axiom of Choice (The Axiom of Choice).

Proof

technique · direct; compare the two forms on a spanning set by the same recursive characterization, then upgrade with the supplier's nondegeneracy

1.1givenF1F2F4algebra

(Tensor adjunction.) Write $A = R (β + γ)$ , $B = R (β) \otimes R (γ)$ and $e = e (β, γ)$ . Reflection fixes each block and gives $(A e \otimes_{B} X)^{ψ} = X^{ψ} \otimes_{B} e A$ , with the same total grading. Thus $T (Ind X, Y) = g d i m (X^{ψ} \otimes_{B} e Y) = T_{B} (X, Res Y)$ . For $X = P ⊠ Q$ , the last tensor factorizes, giving the tensor-product form. Symmetry supplies the adjunction in the other variable. Pull this identity back through $γ$ , using its coproduct compatibility from [F2]. The coefficient of $F_{i} \otimes z$ in $r (y)$ is $e_{i}^{'} y$ : on words this follows by deleting an $i$ and accumulating $q_{i}^{- a_{i j}}$ for each preceding letter in the twisted product, exactly the recursion of [F4]. Weight orthogonality therefore gives $T (γ (F_{i} x), γ (y)) = (1 - q_{i}^{2})^{- 1} T (γ (x), γ (e_{i}^{'} y))$ , with vacuum value $1$ .

1.2givenF3F4algebra

(Abstract side.) $B_{K}$ is characterized by (H), (N), (R) of [F3], and its normalized multiple $B_{L} = C_{ν} B_{K}$ satisfies the generator value $B_{L} (θ_{i}, θ_{j}) = δ_{i j} (1 - q_{i}^{2})^{- 1}$ and the recursion $B_{L} (F_{i} P, Q) = (1 - q_{i}^{2})^{- 1} B_{L} (P, e_{i}^{'} Q)$ by [F4]; the same recursion with the same base value determines every value on word monomials inductively, since peeling the first letter of the first argument and applying the recursion reduces the positive height by one.

2.1step 1.1step 1.2algebra

(Comparison on monomials.) The KLR pairing and $B_{L}$ satisfy the same recursion and the same value at $1$ : on the KLR side this is step 1.1 with $γ (F_{i}) = γ (θ_{i}) = [P_{i, 1}] = [P_{i}]$ and the identification of the restriction class with the coproduct; on the abstract side it is step 1.2. Induction on the height of the weight components therefore gives $(γ (x), γ (y))_{K L R} = B_{L} (x, y)$ on all divided-power monomials and hence on all of $A f$ by bilinearity, using that $γ$ is onto the classes of the divided words by The KLR Grothendieck ring satisfies the divided-power quantum Serre relations. Injectivity of $γ$ is not used: the equality is checked on the spanning monomials.

3.1step 2.1F4F5algebra∎

(Nondegeneracy and generator values.) By [F5] the form $B_{K}$ , hence the unit multiple $B_{L}$ , is nondegenerate on each finite weight component, so the KLR tensor pairing is nondegenerate there as well; the Axiom of Choice enters only through this supplier, and the categorical comparison of steps 1.1, 1.2 and 2.1 is choice-free. The Hom-form identity $(γ (x), γ (y))^{'} = B_{L} (\overline{x}, y)$ follows from $h (P, Q) = T (P^{#}, Q)$ of The KLR Cartan (graded Hom) pairing on the projective Grothendieck group and $γ (\overline{x}) = γ (x)^{#}$ from The KLR Grothendieck ring satisfies the divided-power quantum Serre relations, clause (iii). For the rank-one divided power, $γ (θ_{i}^{(n)}) = [P_{i, n}]$ gives $([P_{i, n}], [P_{i, n}]) = \prod_{a = 1}^{n} (1 - q_{i}^{2 a})^{- 1}$ , matching $B_{L} (θ_{i}^{(n)}, θ_{i}^{(n)})$ by the recursion of step 1.2; for $n = 1$ this is the generator value already recorded.

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The KLR category categorifies the negative quantum half

Statement

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum over any field $k$ . Then the homomorphism $γ : A f \to K_{0} (R)$ of The KLR Grothendieck ring satisfies the divided-power quantum Serre relations is an isomorphism of $Z [q, q^{- 1}]$ -algebras and of $N [I]$ -graded twisted bialgebras: (i) Surjectivity. Every class in $K_{0} (R (β))$ is a $Z [q, q^{- 1}]$ -linear combination of classes of divided-power projectives $[P_{θ}]$ , $θ$ a sequence of divided powers of weight $β$ ; hence $γ$ is onto. (ii) Injectivity. $γ$ is injective, because it preserves the forms by The KLR Cartan pairing is the Lusztig-normalized half form and the source form $B_{L}$ (equivalently $B_{K}$ ) is nondegenerate on each finite weight component. (iii) Compatibility. $γ$ intertwines the multiplications ( $[I n d]$ against the product in $A f$ ), the comultiplications ( $[R e s]$ against $r$ ), the bar involutions and the forms; the free $A$ -rank of $K_{0} (R (β))$ equals the free $A$ -rank of $A f_{β}$ ; no rank or dimension comparison of the infinite-dimensional algebra $R (β)$ itself with $f_{β}$ is asserted. Equivalently, $K_{0} (R)$ is a free $A$ -module with the classes of the indecomposable graded projective $R$ -modules as a shift-orbit basis (Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules), and the divided-power classes span; the triangularly selected classes $[P (Y_{b})]$ form an $A$ -basis, whereas the family of all divided-power words is generally redundant.

Facts & Assumptions

Given: A KLR datum $Q$ over any field $k$ , the algebra $A f$ of the divided-power form of the negative half with its generators $θ_{i}^{(n)}$ of The Lusztig divided-power integral form $U_{A}$ , the homomorphism $γ$ of The KLR Grothendieck ring satisfies the divided-power quantum Serre relations, and the Cartan form comparison of The KLR Cartan pairing is the Lusztig-normalized half form.

[F1]

Triangular basis and simplicity (Maximal KLR tails give a unique induction head and absolute simple modules): for each $β$ there is a finite family of ordered divided-power words $Y_{b}$ with $[P (Y_{b})]$ a triangular $A$ -basis of $K_{0} (R (β))$ with diagonal $1$ , and the graded simples are absolutely simple with all nonzero homogeneous endomorphisms of degree $0$ .

[F2]

Form comparison (The KLR Cartan pairing is the Lusztig-normalized half form): $(γ (x), γ (y))_{K L R} = B_{L} (x, y)$ on each weight component, and $B_{L}$ is a unit multiple of the nondegenerate normalized form $B_{K}$ .

[F3]

Nondegeneracy (The negative-half bilinear form is symmetric, nondegenerate and integral): $B_{K}$ is nondegenerate on each finite weight component; this uses the Axiom of Choice (The Axiom of Choice).

[F4]

Compatibility suppliers: $[R e s]$ is an algebra homomorphism for the twisted product and $γ$ is compatible with coproducts and bar (The KLR Grothendieck ring satisfies the divided-power quantum Serre relations, The Lusztig–Kashiwara form, parameter and coproduct dictionary, The bar involution and the contravariant anti-involution preserve $U_{A}$ ).

Proof

technique · direct; surjectivity by the triangular divided-string basis, injectivity by the preserved nondegenerate form, then compatibility

1.1givenF1algebra

(Surjectivity.) By [F1] the classes $[P (Y_{b})]$ form an $A$ -basis of $K_{0} (R (β))$ for a finite family of ordered divided-power words; each $Y_{b}$ is a word in divided powers of single colours, so its class equals $γ (θ_{i_{1}}^{(n_{1})} \dots θ_{i_{r}}^{(n_{r})})$ by multiplicativity of $γ$ and the definition of $P_{θ}$ . Hence every class in $K_{0} (R (β))$ is an $A$ -linear combination of classes $[P_{θ}]$ of divided-power projectives, and $γ$ is onto.

2.1step 1.1F2F3algebra

(Injectivity.) Suppose $γ (x) = 0$ for some $x \in A f_{ν}$ . By [F2] and [F3], for every $y$ in the same weight component one has $B_{L} (x, y) = (γ (x), γ (y)) = 0$ ; since $B_{L}$ is a unit multiple of the nondegenerate form $B_{K}$ on the finite-dimensional $Q (q)$ -space $A f_{ν} \otimes_{A} Q (q)$ , this forces $x = 0$ . The categorical and form-comparison steps of [F2] are choice-free; the Axiom of Choice is inherited only through the nondegeneracy supplier [F3].

3.1step 1.1step 2.1F1F4algebra∎

(Compatibility and ranks.) The intertwining of multiplications is multiplicativity of $γ$ together with the induction formula of The KLR Grothendieck ring satisfies the divided-power quantum Serre relations; the intertwining of comultiplications is the Mackey comparison of Generic KLR induction and restriction adjunctions preserve projectives against the coproduct $r$ of the negative half; the bar compatibility is $P_{i, n}^{#} = P_{i, n}$ for the normalized divided projectives of Nil-Hecke idempotents and divided-power KLR projectives, and form preservation is [F2]. Since $γ$ is an isomorphism of $A$ -modules by steps 1.1 and 2.1, the free $A$ -rank of $K_{0} (R (β))$ equals that of $A f_{β}$ ; the finite free basis supplied by [F1] transfers through $γ$ to a finite free basis of $A f_{β}$ , and the infinite-dimensional algebra $R (β)$ itself is not compared with $f_{β}$ .

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Self-dual KLR projectives have positive first-block divided-string leading terms

Statement

For any graded KLR datum in this page and any field $k$ , each finite-dimensional graded simple shift orbit has a unique representative $S_{b}$ self-dual for $D (M)_{d} = {Hom}_{k} (M_{- d}, k)$ with action $(a f) (m) = f (ψ (a) m)$ . Its projective cover $P_{b}$ , normalized by ${Hom}^{0} (P_{b}, S_{b}) = k$ , satisfies $P_{b}^{#} = P_{b}$ for $P^{#} = {HOM}_{R} (P, R)^{ψ}$ . These are unique normalizations. Induction of two self-dual finite projectives is self-dual. If $ϵ_{i}^{L} (S_{b}) = r > 0$ and $Δ_{i}^{(L, r)} S_{b} = L (i^{r}) ⊠ S_{a}$ with $ϵ_{i}^{L} (S_{a}) = 0$ , normalized as above, then $[P_{i, r}] [P_{a}] = [P_{b}] + \sum_{ϵ_{i}^{L} (S_{c}) > r} m_{c} (q) [P_{c}],$ with $m_{c} \in N [q, q^{- 1}]$ each invariant under $q \mapsto q^{- 1}$ . Under $γ$ the integral LEFT divided-string ideal $I_{(i, \geq n)} = \sum_{r \geq n} F_{i}^{(r)} U_{A}^{-}$ is the $A$ -span of $[P_{b}]$ with $ϵ_{i}^{L} (S_{b}) \geq n$ . Consequently, if the normalized projective classes are known to be a SIGNED lower-global set, this leading coefficient $1$ identifies them with the POSITIVE lower-global set using the general adapted LEFT-string recursion; no finite-ADE canonical-comparison theorem is needed. For the conclusions compared with the quantum half and the conditional lower-global/LEFT-string recursion, carry the inherited Axiom of Choice of those quantum suppliers; the generic finite module duality/parity/Frobenius arguments themselves are choice-free.

Facts & Assumptions

Given: A graded KLR datum over any field $k$ , the graded simple modules and projective covers of Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules with the maximal-tail triangular basis and absolute simplicity of Maximal KLR tails give a unique induction head and absolute simple modules, the duality $P^{#}$ and pairing of The KLR Cartan (graded Hom) pairing on the projective Grothendieck group, and the isomorphism $γ$ of The KLR category categorifies the negative quantum half.

[F1]

Maximal-tail basis (Maximal KLR tails give a unique induction head and absolute simple modules): there is a finite triangular $A$ -matrix of divided-idempotent Homs against simples with diagonal $1$ , the simples are absolutely simple, and the selected divided words $Y_{b}$ give a triangular $A$ -basis of $K_{0} (R (β))$ ; the identical assertions holds for maximal FIRST-colour blocks.

[F2]

Divided powers (Nil-Hecke idempotents and divided-power KLR projectives): $P_{i, n}$ is the unique indecomposable projective summand of $R (n α_{i})$ up to shift and $P_{i, n}^{#} = P_{i, n}$ ; $[P_{i, n}] = γ (θ_{i}^{(n)})$ .

[F3]

Categorification (The KLR category categorifies the negative quantum half): $γ$ is an isomorphism of $A$ -algebras intertwining induction products with the algebra product, restriction with the coproduct, and the bar involutions, with both sides free over $A$ componentwise.

[F4]

Global suppliers (Bar invariance, congruence and uniqueness of the lower global basis, The highest weight and negative half integral crystal triples are balanced): the lower global basis of the negative half and its integral balanced globalization, and the adapted LEFT-string ideal decomposition and normalized transition of clause (v) of the balanced theorem: if ${\tilde{e}}_{i} a = 0$ , then $F_{i}^{(r)} G (a) - G ({\tilde{f}}_{i}^{r} a) \in I_{(i, \geq r + 1)}$ , are available under the stated hypotheses.

Proof

technique · direct; normalize dualities by word characters, compute the first-block expansion, then compare with the integral string ideals and the global recursion

1.1givenF1algebra

(Self-dual simples.) The maximal-tail supplier [F1] gives a triangular divided-idempotent/simple Hom matrix with diagonal $1$ over $A = Z [q, q^{- 1}]$ , which remains invertible at $q = 1$ . Ordinary dimensions of divided-word corners are determined by ordinary word dimensions: in each constant-colour block the nil-Hecke matrix presentation makes the polynomial column simple of rank $r!$ and the divided idempotent a rank-one matrix projection, so $\dim (e_{div} M) = \dim (e_{expanded} M) / \prod_{r} r!$ for every finite module $M$ , including nonsemisimple central quotients. Hence ordinary word characters separate simple shift orbits. The vector duality $D (M)_{d} = {Hom}_{k} (M_{- d}, k)$ preserves simple modules and fixes ordinary word dimensions because $ψ$ fixes the word idempotents, so $D S ≅ S {s}$ for some integer $s$ .

2.1step 1.1algebra

(Evenness and normalization.) Choose a total colour order and set $p (ν) = \sum_{a < b, ν_{a} > ν_{b}} b_{ν_{a} ν_{b}} m o d 2$ . Each adjacent unequal swap changes $p$ by its crossing degree modulo $2$ , and equal crossings and dots have even degree, so the PBW basis puts $e (η) R e (ν)$ in degrees congruent to $p (η) - p (ν)$ ; a graded module therefore splits into the two submodules on which $d - p (ν)$ is fixed modulo $2$ on $e (ν) M_{d}$ , and a simple uses one of them. Since $- d - p (ν) \equiv d - p (ν) (m o d 2)$ , $D S$ has the same sector as $S$ , so the shift $s$ above is even; replacing $S$ by $S {s / 2}$ gives $D S = S$ , and uniqueness follows because a nonzero finite-dimensional graded module is not isomorphic to a nonzero shift of itself (compare its finite least and greatest degrees).

3.1step 2.1F2algebra

(Self-dual covers and induction.) For a finite projective $P$ and finite module $M$ , evaluation gives $Hom (P^{#}, D M) = D Hom (P, M)$ with ordinary homogeneous degrees: verify on $P = R e (ν) {a}$ , where both sides are the dual of the finite-dimensional space $e (ν) M {- a}$ (Nil-Hecke idempotents and divided-power KLR projectives), and restrict to a finite summand. The duality $^{#}$ is an involution on finite projectives, so $P_{b}^{#}$ is the same normalized indecomposable projective cover of $D S_{b} = S_{b}$ , giving $P_{b}^{#} = P_{b}$ ; a nonzero projective is bounded below, so the normalization is unique. If $M$ is a finite projective over the separated block algebra $B$ , then ${Hom}_{R} (R e \otimes_{B} M, R) = {Hom}_{B} (M, e R) = {Hom}_{B} (M, B) \otimes_{B} e R$ by evaluation on finite free modules and summands, and twisting by $ψ$ identifies this with $R e \otimes_{B} M^{#}$ without reversing the horizontal block order; for $M = P_{a} ⊠ P_{i, r}$ this proves that $I n d M$ is self-dual. Restriction commutes with $D$ because $ψ$ fixes the defining block idempotent, so the maximal FIRST-block restriction of the normalized $S_{b}$ is exactly the normalized $L (i^{r}) ⊠ S_{a}$ with no residual shift: that tensor is $D$ -self-dual and any extra shift would contradict finite support.

4.1step 3.1F1F2algebra

(Leading coefficient $1$ .) Use the exact tensor/Res adjunction of Generic KLR induction and restriction adjunctions preserve projectives and the maximal-tail lemma [F1] to compute $Hom (I n d (P_{i, r} ⊠ P_{a}), S_{c})$ : it vanishes if $ϵ_{i}^{L} (S_{c}) < r$ ; if $ϵ_{i}^{L} (S_{c}) = r$ , the maximal restriction is $L (i^{r}) ⊠ S_{a^{'}}$ and the Hom is zero unless $a^{'} = a$ , when it is exactly $k$ in degree $0$ ; the unique-head lemma makes $c = b$ in that case. Since Hom from a projective detects the multiplicity polynomial of each summand in the decomposition, the induced projective has exactly $[P_{b}]$ with coefficient $1$ at tail $r$ and all other summands at strictly higher tail, giving the displayed expansion. Self-duality of the induction and of each normalized projective forces every multiplicity polynomial to be bar invariant, and finite decompositions give nonnegative coefficients.

5.1step 4.1F3algebra

(Integral string ideals.) For every $n$ , induction with a divided block of length $r \geq n$ has no summands of tail $< n$ by the same Hom computation of step 4.1, so $γ (I_{(i, \geq n)})$ is contained in the span of the corresponding covers. Conversely descend on the finite possible tail $r$ (bounded by $β_{i}$ ): the coefficient- $1$ expansion expresses every $[P_{b}]$ of tail $r$ as $[P_{i, r}] [P_{a}]$ minus classes of strictly higher tail, which lie in $I_{(i, \geq r)}$ already; this proves the reverse inclusion integrally, using the sum of all divided images and not a Gaussian-unit identification with a single ordinary power ideal.

6.1step 5.1F3F4algebra∎

(Signs.) Suppose $[P_{b}] = γ ({s i g n}_{b} G (b^{'}))$ has been established by a separate norm/geometric argument. The ideal equality of step 5.1 and the adapted LEFT-string lower-global basis [F4] identify the tail partitions of these signed vectors with the LEFT string partitions; at the exact LEFT-string quotient the lower-global recursion has coefficient $+ 1$ for $F_{i}^{(r)} G (a^{'})$ when $a^{'}$ has LEFT string $0$ . Induction on root height fixes ${s i g n}_{a} = + 1$ , and comparing the projective coefficient- $1$ expansion with this quotient fixes ${s i g n}_{b} = + 1$ ; the unit/vacuum starts the induction. The reversal supplier used elsewhere promises only signed star preservation, so an unsigned right recursion is deliberately not inferred here. The Axiom of Choice enters only through the inherited quantum-half/global suppliers [F4] and The KLR category categorifies the negative quantum half (The Axiom of Choice); no additional simultaneous module-basis choice is used, and the finite duality, parity, Frobenius and first-block multiplicity arguments above are choice-free.

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Self-dual indecomposable KLR projectives give the lower global basis (symmetric type)

Statement

Assume the Cartan datum of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum has symmetric Cartan matrix $a_{i j} = a_{j i}$ and uses the quiver normalization $d_{i} = 1$ for every $i$ , the field $k$ has characteristic $0$ , and the KLR parameters are the quiver-geometric ones (the specialization attached to a finite quiver without loops, with arbitrary arrow multiplicities $h_{i j}$ and $a_{i j} = - (h_{i j} + h_{j i})$ : $Q_{i j} (u, v) = (- 1)^{h_{i j}} (u - v)^{h_{i j} + h_{j i}}$ for $i \neq j$ , in the fixed native crossing-sign convention; arbitrary parameter changes, including a global replacement by $u + v$ , require their own proved gauge equivalence). Then, normalizing each indecomposable finitely generated graded projective $R (β)$ -module to be self-dual for the contravariant graded projective duality $P^{#} = {HOM}_{R (β)} (P, R (β))^{ψ}$ ( $P ≅ P^{#}$ after the unique normalizing shift), the map $b \mapsto [P_{b}]$ is a bijection between the lower global basis $B (\infty)$ of the negative half (The lower global basis of $U_{q}^{-}$ and of a highest weight module, Bar invariance, congruence and uniqueness of the lower global basis) and the shift-orbit basis of $K_{0} (R)$ by self-dual indecomposable projectives; under the isomorphism $γ$ of The KLR category categorifies the negative quantum half one has $γ (G (b)) = [P_{b}]$ for the lower global lift $G (b)$ , and the identification is compatible with the based quotient of The based quotient: projection of the lower global basis to a highest weight module. No extension to arbitrary KLR parameters or to non-symmetric Cartan data is asserted here; any such extension requires its own parameter-change invariance theorem.

Facts & Assumptions

Given: The symmetric quiver-normalized Cartan datum with $d_{i} = 1$ , a characteristic-zero field $k$ , the quiver-geometric parameters $Q_{i j} (u, v) = (- 1)^{h_{i j}} (u - v)^{h_{i j} + h_{j i}}$ , the categorical form of The KLR Cartan (graded Hom) pairing on the projective Grothendieck group, the isomorphism $γ$ of The KLR category categorifies the negative quantum half, and the QG-5 global-basis machinery of The lower global basis of $U_{q}^{-}$ and of a highest weight module, Bar invariance, congruence and uniqueness of the lower global basis, The based quotient: projection of the lower global basis to a highest weight module, Crystal lattice polarizations are perfect at $q = 0$ , Integral residues of the negative half: freeness, the crystal basis and the form characterization, The Lusztig–Kashiwara form, parameter and coproduct dictionary and The highest weight and negative half integral crystal triples are balanced.

Choice. Assume the inherited Axiom of Choice of the geometric and quantum suppliers; the finite presented-algebra comparison and finite graded Yoneda argument below introduce no additional choice.

[F1]

Categorical input (The KLR category categorifies the negative quantum half, The KLR Cartan (graded Hom) pairing on the projective Grothendieck group): $γ : A f \to K_{0} (R)$ is an isomorphism of $A$ -algebras and twisted bialgebras intertwining the forms, and every indecomposable finite graded projective shift orbit has a unique normalized self-dual representative by Self-dual KLR projectives have positive first-block divided-string leading terms, with the pairing of The KLR Cartan (graded Hom) pairing on the projective Grothendieck group.

[F2]

Positive first-string supplier (Self-dual KLR projectives have positive first-block divided-string leading terms): normalized self-dual projective covers satisfy the coefficient- $1$ FIRST-block expansion, the integral LEFT divided-string ideal equality, and the positive FIRST/LEFT global comparison; its inherited Axiom of Choice (The Axiom of Choice) is carried here.

[F3]

Actual geometric realization (Quiver full-flag incidence varieties and signed geometric KLR generators, Signed KLR algebra is the equivariant Ext algebra of quiver-flag complexes, Quiver-flag convolution generators satisfy the signed KLR relations, Relative-position incidence strata give a faithful geometric PBW basis): the intrinsic presentation has the full distinct-color adjacent dot slides, native equal-color mixed and braid signs, and a faithful action generated by the native polynomial operators on summands with unit degree $- d_{ν}$ ; the Ext product is $a * b = b [n] \circ a$ for $a$ of degree $n$ , with source-first corner $e_{η} R e_{ν} = {Ext}_{G} (L_{η}, L_{ν})$ .

[F4]

Actual semisimplicity (Full quiver-flag pushforwards are semisimple in characteristic zero): for every finite loopless quiver with arbitrary finite arrow multiplicities and every characteristic-zero field, $L_{ν} = R π_{ν!} k [d_{ν}]$ and their finite sum are finite direct sums of shifts of simple objects in the actual equivariant perverse heart of Middle perverse t structure glues and has simple intersection complexes. That bounded heart gives $Hom (T, T [n]) = 0$ for $n < 0$ and division $End (T)$ for simple $T$ .

[F5]

Projective covers (Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules, Maximal KLR tails give a unique induction head and absolute simple modules): graded indecomposable finite projectives are projective covers of finite-dimensional graded simples, and those simples are absolutely simple over $k$ .

[F6]

Actual splitting (Cartesian simplicial sheaves model actual equivariant derived Hom): the bounded equivariant constructible category is triangulated and idempotent complete.

Proof

technique · direct; build the geometric realization, run the graded-Yoneda argument, then globalize and fix the signs with the positive first-string supplier

1.1givenF3algebra

(The graded algebra comparison.) Set $P_{i j} (u, v) = (v - u)^{h_{i j}}$ for $i \neq j$ . These are nonzero, including when $h_{i j} = 0$ , and $P_{i j} (u, v) P_{j i} (v, u) = (- 1)^{h_{i j}} (u - v)^{h_{i j} + h_{j i}} = Q_{i j} (u, v)$ . Thus Faithful polynomial representation of a KLR algebra gives a faithful action of $R (β; Q)$ by projections, dots and unequal crossings $(x_{r} - x_{r + 1})^{h_{i j}} s_{r}$ , with equal crossings $(1 - s_{r}) / (x_{r} - x_{r + 1})$ . The actual geometric action of [F3] uses exactly those operators. In both actions give the unit of the $ν$ polynomial summand degree $- d_{ν}$ : the incidence formula $d_{ν} = \sum_{i} β_{i} (β_{i} - 1) / 2 + \sum_{a > b} h_{ν_{a} ν_{b}}$ gives $d_{s_{r} ν} - d_{ν} = h_{i j} - h_{j i}$ , so the unequal crossing has degree $2 h_{i j} + d_{ν} - d_{s_{r} ν} = h_{i j} + h_{j i}$ ; equal crossings have degree $- 2$ and dots degree $2$ . Both faithful images are therefore the identical graded subalgebra generated by these operators, yielding $R (β; Q) ≅ S_{Γ} (β) ≅ {Ext}_{G}^{∙} (L, L)$ for $L = ⨁_{ν} L_{ν}$ . This is also the direct comparison of the identical specialized presentations: all adjacent unequal dot slides and the native mixed and braid signs are present in [F3]. The equal geometric crossing is minus its standard complex Borel–Moore class, so no sign change is implicit.

2.1F4step 1.1

(The actual normalized complex.) By [F4], the same $L = ⨁_{ν} R π_{ν!} k [d_{ν}]$ of step 1.1 is a finite direct sum of shifts of simple actual equivariant perverse objects. The normalization $[d_{ν}]$ has already been included in $L_{ν}$ ; no further shift of $L$ is made. This uses the proved semisimplicity clause in the complete loopless multiedge scope, with no semismallness or finite-ADE restriction.

2.2F3F6step 1.1algebra

(Graded Yoneda.) Use the shifted opposite-composition product of [F3] and define $Y_{T} = {Ext}_{G}^{∙} (L, T)$ as a graded left $R$ -module: for homogeneous $a : L \to L [n]$ and $y : L \to T [m]$ , set $a \cdot y = y [n] \circ a$ , so $(a * b) \cdot y = a \cdot (b \cdot y)$ . On $T = L$ this is the regular left module. A homogeneous map $f : T \to U [t]$ acts by postcomposition $y \mapsto f [m] \circ y$ of degree $t$ . For $T = U = L$ every graded $R$ -linear map $R \to R$ is right multiplication by its value at $1$ , hence is induced uniquely by the corresponding $f$ ; extend this identification to finite sums and shifts and then to idempotent summands. Since the actual category is idempotent complete and all finite graded projectives are summands of finite sums of shifts of $R$ , this gives a fully faithful equivalence from the additive category generated by shifts of $L$ onto finite graded projectives, with ${End}_{R}^{∙} (Y_{T}) ≅ {Ext}_{G}^{∙} (T, T)$ using ordinary categorical composition on both endomorphism rings.

3.1F4F5step 2.1step 2.2algebra

(Indecomposables and scalar degree zero.) Each simple perverse summand $T$ in step 2.1 belongs to that additive category up to shift. By [F4] and step 2.2 its projective $Y_{T}$ has no negative-degree endomorphisms and its degree-zero endomorphism ring is a division algebra, hence it is graded indecomposable. These exhaust the indecomposable projective shift orbits: every projective is a summand of a finite sum of the $Y_{T}$ and their shifts, and [F5] gives the finite graded Krull–Schmidt decomposition. Distinct simple perverse objects yield distinct orbits, since full faithfulness would otherwise give $T ≅ U [t]$ , and perverse concentration forces $t = 0$ . Finally the simple head $S$ of $Y_{T}$ has ${End}_{R}^{0} (S) = k$ by [F5]. Every degree-zero endomorphism preserves the unique maximal graded submodule, so descent to the head is a unital $k$ -algebra homomorphism ${End}_{R}^{0} (Y_{T}) \to k$ ; its kernel is zero because its source is a division algebra, and its image contains all scalars, so ${End}_{R}^{0} (Y_{T}) = k$ .

4.1step 3.1F1F2algebra

(Self-dual normalization, norm one and signed globalization.) By [F1] every indecomposable projective shift orbit has a unique $^{#}$ self-dual representative $P_{b}$ ; put $x_{b} = γ^{- 1} [P_{b}]$ , which is integral and bar fixed. The exact Hom/tensor comparison for a self-dual $P$ gives $B_{L} (x_{b}, x_{b}) = h (P_{b}, P_{b})$ , a nonnegative integer power series in $q$ congruent to $1$ modulo $q$ , since the geometric nonnegative-End property is unchanged by grading shifts. Dividing by $C_{β}$ (which is $1$ modulo $q$ ) gives a value $B_{K} (x_{b}, x_{b})$ congruent to $1$ modulo $q$ with integer coefficients, so the general half norm criterion places $x_{b}$ in the $q^{0}$ crystal lattice; being bar fixed, it also lies in the actual bar chart, and the integral balanced globalization writes $x_{b}$ as a finite $Z$ -linear combination of lower global vectors. Their residue Gram matrix is the identity, so norm residue $1$ forces the sum of coefficient squares to be $1$ : $x_{b}$ is exactly plus or minus one lower global vector. Distinct projective shift orbits are independent and have the same finite weight cardinality as the global basis by the integral $K_{0}$ isomorphism, so they exhaust this signed set.

5.1step 4.1F2algebra∎

(Positivity and the based quotient.) The positive divided-string leading-term supplier [F2] removes every minus sign using its integral LEFT-string ideal equality, its coefficient- $1$ projective adjunction expansion, and the general adapted LEFT global recursion (carried with its inherited Axiom of Choice, The Axiom of Choice); batch 8's finite-ADE canonical-comparison item is not used. Consequently the sign in step 4.1 is $+ 1$ , giving $γ (G (b)) = [P_{b}]$ for every lower global vector $G (b)$ and establishing the bijection of the statement. The compatibility with the based quotient is the general zero/survival projection theorem of the QG-5 machinery.

5 · Examples, counterexamples and false statements

None yet.