Affine zigzag algebras and imaginary strata for KLR algebras

Kleshchev, Alexander; Muth, Robert

doi:10.1090/tran/7464

Cited by 14 publications

(13 citation statements)

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“…The main result of Section 6 is Theorem 6.14, which gives a partial description of the algebra e δ d R dδ e δ d . A similar (but more explicit) result has independently been obtained by Kleshchev and Muth [22] for all KLR algebras of untwisted affine ADE types. More precisely, Theorem 6.14 can be deduced from [22,Theorem 5.9], which describes a 2 For any fixed i ∈ I ∅,1 , Kleshchev and Muth find a certain element of R de , denoted by σr + c in [21, (4.2.3)], such that the image of this element in R dδ multiplied by our Θ(e(i) ⊗d ⊗1) is equal to Θ(e(i) ⊗d ⊗sr).…”

Section: 3supporting

confidence: 78%

“…A similar (but more explicit) result has independently been obtained by Kleshchev and Muth [22] for all KLR algebras of untwisted affine ADE types. More precisely, Theorem 6.14 can be deduced from [22,Theorem 5.9], which describes a 2 For any fixed i ∈ I ∅,1 , Kleshchev and Muth find a certain element of R de , denoted by σr + c in [21, (4.2.3)], such that the image of this element in R dδ multiplied by our Θ(e(i) ⊗d ⊗1) is equal to Θ(e(i) ⊗d ⊗sr). certain idempotent truncation of e δ d R dδ e δ d as an affine zigzag algebra and is proved using explicit diagrammatic computations, which are generally avoided below.…”

Section: 3supporting

confidence: 78%

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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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Section: 3supporting

confidence: 78%

Section: 3supporting

confidence: 78%

RoCK blocks, wreath products and KLR algebras

Evseev

2016

Math. Ann.

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“…In other cases, Theorem 6.11 seems to be new. In particular, as noted in the introduction, when F is a symmetric algebra, concentrated in even parity, A n (F ) is the affinized symmetric algebra considered by Kleshchev and Muth [KM,§3]. Under these additional assumptions, those authors prove that the elements given in Theorem 6.11 are a spanning set and then conjecture that they are a basis [KM,Conj.…”

Section: Proposition 62 Suppose C ∈ Fmentioning

confidence: 99%

Affine Wreath Product Algebras

Savage

2018

International Mathematics Research Notices

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We study the structure and representation theory of affine wreath product algebras and their cyclotomic quotients. These algebras, which appear naturally in Heisenberg categorification, simultaneously unify and generalize many important algebras appearing in the literature. In particular, special cases include degenerate affine Hecke algebras, affine Sergeev algebras (degenerate affine Hecke-Clifford algebras), and wreath Hecke algebras. In some cases, specializing the results of the current paper recovers known results, but with unified and simplified proofs. In other cases, we obtain new results, including proofs of two open conjectures of Kleshchev and Muth.

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“…, z n ] Sn , but in the imaginary case B nδ is not so easy to understand. In the sequel paper [21], we reveal a connection between B nδ and affine zigzag algebras related to the 'finite zigzag algebras' of [11] of the underlying finite Lie type.…”

Section: Introductionmentioning

confidence: 99%