RoCK blocks, wreath products and KLR algebras

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 $${\mathfrak {S}}_n$$ defined over a field F of characteristic e is Morita equivalent to the principal block of the wreath product $$\mathfrak S_e \wr \mathfrak S_d$$ . 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 $$F{\mathfrak {S}}_n$$ 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 Iwahori–Hecke algebra at a root of unity defined over an arbitrary field.