Rock blocks

Turner, Will

doi:10.1090/s0065-9266-09-00562-6

Cited by 24 publications

(25 citation statements)

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“…Considering the long history of the theory of symmetric groups, it is quite surprising that one can define such a non-trivial grading on the symmetric group algebras, which has been conjectured to exist for some time ( [Rou2,Tur]), only after the discovery of Khovanov-Lauda-Rouquier algebras. For the Dynkin diagrams of affine type A or of type A ∞ , the Khovanov-Lauda-Rouquier algebras (resp.…”

Section: Introductionmentioning

confidence: 99%

Quiver Hecke superalgebras

Kang

Kashiwara

Tsuchioka

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We introduce a new family of superalgebras which should be considered as a super version of the Khovanov-Lauda-Rouquier algebras. Let I be the set of vertices of a Dynkin diagram with a decomposition I = I even ⊔ I odd . To this data, we associate a family of graded superalgebras R n , the quiver Hecke superalgebras. When I odd = ∅, these algebras are nothing but the usual Khovanov-Lauda-Rouquier algebras. We then define another family of graded superalgebras RC n , the quiver Hecke-Clifford superalgebras, and show that the superalgebras R n and RC n are weakly Morita superequivalent to each other. Moreover, we prove that the affine Hecke-Clifford superalgebras, as well as their degenerate version, the affine Sergeev superalgebras, are isomorphic to quiver Hecke-Clifford superalgebras RC n after a completion.2000 Mathematics Subject Classification. Primary 81R50, Secondary 20C08.

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Section: Introductionmentioning

confidence: 99%

Quiver Hecke superalgebras

Kang

Kashiwara

Tsuchioka

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…The cyclotomic Hecke algebras of type A have a uniform description but, historically, they have been studied either as Ariki-Koike algebras (v = 1), or as degenerate Ariki-Koike algebras (v = 1). The existence of gradings on Hecke algebras, at least in the "abelian defect case", was predicted by Rouquier [120,Remark 3.11] and Turner [130].…”

Section: Introductionmentioning

confidence: 95%

CYCLOTOMIC QUIVER HECKE ALGEBRAS OF TYPE A

Mathas

2015

Modular Representation Theory of Finite and p-Adic Groups

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Contents n 1.6. Semisimple cyclotomic Hecke algebras of type A 1.7. Gram determinants and the Jantzen sum formula 1.8. The blocks of H F n 2. Cyclotomic quiver Hecke algebras of type A 2.1. Graded algebras 2.2. Cyclotomic quiver Hecke algebras 2.3. Nilpotence and small representations 2.4. Semisimple KLR algebras 2.5. The nil-Hecke algebra 3. Isomorphisms, Specht modules and categorification 3.1. The Graded Isomorphism Theorem 3.2. Graded Specht modules 3.3. Blocks and dual Specht modules 3.4. Induction and restriction 3.5. Grading Ariki's Categorification Theorem 3.6. Homogeneous Garnir relations 3.7. Graded adjustment matrices 4. Seminormal bases and the KLR grading 4.1. Gram determinants and graded dimensions 4.2. A deformation of the KLR grading 4.3. A distinguished homogeneous basis 4.4. A simple conjecture References

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“…After a certain relabelling, this matrix may be seen to be identical to the decomposition matrix of H ∅,1 ≀ S d . When ξ = 1, the decomposition matrix of a RoCK block H ρ,d was determined by Turner [33,Theorem 132] and was shown to coincide with that of H ∅,1 ≀ S d by Paget [29,Theorem 3.4], again, after a certain relabelling. These results are closely related to Theorem 1.1 but are not directly implied by it, as we do not describe explicitly the H ∅,1 ≀ S d -modules which are the images of Specht and simple modules of H ρ,d under the composition of the Morita equivalence of Theorem 1.1 and the functor F.…”

Section: Introductionmentioning

confidence: 99%

“…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: 99%

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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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Rock blocks

Cited by 24 publications

References 31 publications

Quiver Hecke superalgebras

Quiver Hecke superalgebras

CYCLOTOMIC QUIVER HECKE ALGEBRAS OF TYPE A

RoCK blocks, wreath products and KLR algebras

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