RoCK blocks, wreath products and KLR algebras

Abstract

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...

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

“…For any i ∈ I, Define H(r, s, i) ⊂ Z × Z be the e-hook with arm length i and vertex (x(r, s, i), y(r, s, i)). The following lemma is a refinement of [CK,Lemma 4] and [Ev,Lemma 4.3].…”

“…We will need the following weak version of the Mackey Theorem for KLR algebras, see [Ev,Proposition 3.7] or the proof of [KL,Proposition 2.18]:…”

“…We now review and develop some results from [Ev,Section 4]. Throughout the subsection, we fix d ∈ Z >0 and a d-Rouquier core ρ of residue κ.…”

“…Indeed, it is straightforward to check that both sides are 0 when τ = ∅, since b 0 (∅) = · · · = b m−1 (∅) = b m (∅) + 1 = · · · = b e−1 (∅) + 1. Furthermore, adding a box of residue i ∈ I to τ changes both sides by e − 1 if i = κ and by −1 if i = κ (for the right-hand side, consult [Ev,Lemma 4.2]). The claim is proved.…”

“…Denoting by τ the translation of Z × Z which maps (1, 1) to v, we have a bijection Std(µ \ ρ, (l k ) +κ ) ∼ −→ Std(ν, l k ) given by t → s where s(u) = t(τ (u)) for all u ∈ ν (and similarly for l j ). Moreover, we have deg(s) = deg(t) by [Ev,(4.6…”