Blocks of symmetric groups, semicuspidal KLR algebras and zigzag Schur-Weyl duality

Evseev, Anton; Kleshchev, Alexander

doi:10.4007/annals.2018.188.2.2

Cited by 18 publications

(14 citation statements)

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“…In particular, B δ ∼ = [z] ⊗ Z. Theorem A appears in the body of the paper as Theorem 6.16 and Corollary 6.17. We note that Theorem A has been used in a crucial way in the recent proof of Turners conjecture on RoCK blocks of symmetric groups [8], [7].…”

Section: Introductionmentioning

confidence: 99%

Affine zigzag algebras and imaginary strata for KLR algebras

Kleshchev

Muth

2018

Trans. Amer. Math. Soc.

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KLR algebras of affine ADE types are known to be properly stratified if the characteristic of the ground field is greater than some explicit bound. Understanding the strata of this stratification reduces to semicuspidal cases, which split into real and imaginary subcases. Real semicuspidal strata are well-understood. We show that the smallest imaginary stratum is Morita equivalent to Huerfano-Khovanov's zigzag algebra tensored with a polynomial algebra in one variable. We introduce affine zigzag algebras and prove that these are Morita equivalent to arbitrary imaginary strata if the characteristic of the ground field is greater than the bound mentioned above.

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Section: Introductionmentioning

confidence: 99%

Affine zigzag algebras and imaginary strata for KLR algebras

Kleshchev

Muth

2018

Trans. Amer. Math. Soc.

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“…In general, simple imaginary semicuspidal R nδ -modules are not isomorphic to skew Specht modules, but do appear to arise as heads (or socles) of imaginary semicuspidal skew Specht modules. For a balanced convex preorder, evidence for this assertion appears in [7], which relates some semicuspidal R nδ -modules to RoCK blocks of Hecke algebras via Morita equivalence. For arbitrary convex preorders, we expect a similar connection to hold with blocks which are Scopes equivalent to RoCK blocks.…”

Section: An Application To Representation Theory Of Klr Algebrasmentioning

confidence: 99%

Cuspidal ribbon tableaux in affine type A

Abbasian¹,

Lena²,

Muth³

et al. 2020

Preprint

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For any convex preorder on the set of positive roots of affine type A, we classify and construct all associated cuspidal and semicuspidal skew shapes. These combinatorial objects correspond to cuspidal and semicuspidal skew Specht modules for the Khovanov-Lauda-Rouquier algebra of affine type A. Cuspidal skew shapes are ribbons, and we show that every skew shape has a unique ordered tiling by cuspidal ribbons. This tiling data provides an upper bound, in the bilexicographic order on Kostant partitions, for labels of simple factors of Specht modules.

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“…In addition to conjecturing the statement of Theorem 1.1, Turner [33] has constructed two remarkable algebras that he conjectured to be Morita equivalent respectively to the whole RoCK block H ρ,d and to a RoCK block of a ξ-Schur algebra (see [33,Conjectures 165 and 178] respectively). After the present paper was submitted, the first of these conjectures was proved in [13] using results contained here.…”

Section: Introductionmentioning

confidence: 95%

RoCK blocks, wreath products and KLR algebras

Evseev

2016

Math. Ann.

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Abstract. We consider RoCK (or Rouquier) blocks of symmetric groups and Hecke algebras at roots of unity. We prove a conjecture of Turner asserting that a certain idempotent truncation of a RoCK block of weight d of a symmetric group Sn defined over a field F of characteristic e is Morita equivalent to the principal block of the wreath product Se ≀ S d . This generalises a theorem of Chuang and Kessar that applies to RoCK blocks with abelian defect groups. Our proof relies crucially on an isomorphism between F Sn and a cyclotomic Khovanov-Lauda-Rouquier algebra, and the Morita equivalence we produce is that of graded algebras. We also prove the analogous result for an IwahoriHecke algebra at a root of unity defined over an arbitrary field.1. Introduction 1.1. The main result. Let ξ be a fixed element of an arbitrary field F . We assume that there exists an integer e ≥ 2 such that 1 + ξ + · · · + ξ e−1 = 0 and let e be the smallest such integer (the quantum characteristic of ξ). We fix e, F and ξ throughout the paper.For an integral domain O, an invertible element ξ ∈ O and an integer n ≥ 0, the Iwahori-Hecke algebra H n (O, ξ) is the O-algebra defined by the generators T 1 , . . . , T n−1 subject to the relations (T r − ξ)(T r + 1) = 0 for 1 ≤ r < n, (1.1)for 1 ≤ r, s < n such that |r − s| > 1. (1.3) Throughout, we write H n = H n (F, ξ). The algebra H n is cellular, and hence F is necessarily a splitting field for this algebra (see e.g. [26, Theorem 3.20 ]).It is well known that the blocks of H n are parameterised by the setwhere Par is the set of all partitions. We write b ρ,d for the block idempotent of H n corresponding to (ρ, d) ∈ Bl e (n), andH n denotes the corresponding block (see Section 2 for details). Representation theory of RoCK (or Rouquier ) blocks of H n (see Definition 2.1) is much more tractable than that of blocks H ρ,d in general. By a fundamental result of Chuang and Rouquier [8, Section 7], for any d ≥ 0 and any two ecores ρ (1) and ρ (2) , the algebras H ρ (1) ,d and H ρ (2) ,d are derived equivalent. Consequently, in order to understand the structure of an arbitrary block H ρ,d up to derived equivalence, it suffices to give a description of the structure of each RoCK block up to derived equivalence. If ξ = 1, then e = char F is necessarily prime and H n ∼ = F S n , where S n denotes the symmetric group on n letters. Chuang and Kessar [6] proved that, when ξ = 1 and d < char F = e, a RoCK block H ρ,d is Morita equivalent to the wreath product H ∅,1 ≀ S d . Note that here H ∅,1 is the principal block of F S e and that the result of [6] applies precisely to RoCK blocks of symmetric groups with abelian defect. In fact, the aforementioned theorems of Chuang-Rouquier and Chuang-Kessar are stronger, as they hold with F replaced by an appropriate discrete valuation ring. More precisely, for any integers 0 ≤ m ≤ n, view H m as a subalgebra of H n via the embedding T j → T j for 1 ≤ j < m. For any e-core ρ and d ≥ 0, defineClearly, the factors in this product commute pairwise, so f ρ,d is an idempo...

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Blocks of symmetric groups, semicuspidal KLR algebras and zigzag Schur-Weyl duality

Cited by 18 publications

References 35 publications

Affine zigzag algebras and imaginary strata for KLR algebras

Affine zigzag algebras and imaginary strata for KLR algebras

Cuspidal ribbon tableaux in affine type A

RoCK blocks, wreath products and KLR algebras

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