RoCK blocks, wreath products and KLR algebras

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.