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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.

Depends on

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Sources

Raphael Rouquier, 2-Kac-Moody Algebras (standard reference, not scraped)
Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups I, Representation Theory 13 (2009) 309-347, arXiv:0803.4121v2 (standard reference, not scraped)
Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups II, Transactions of the AMS 363 (2011) 2685-2700, arXiv:0804.2080v1 (standard reference, not scraped)
Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, Bulletin of the AMS 47 (2010) 419-481 (survey), arXiv:0909.4844 (standard reference, not scraped)
Seok-Jin Kang and Masaki Kashiwara, Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras, arXiv:1102.4677v4 (42 pages) (standard reference, not scraped)