RoCK blocks, wreath products and KLR algebras

TL;DR

This paper proves a Morita equivalence between certain blocks of symmetric groups and wreath products, extending previous results and using KLR algebra isomorphisms, applicable over arbitrary fields.

Contribution

It establishes a new Morita equivalence for RoCK blocks of symmetric groups and Hecke algebras at roots of unity, generalizing prior theorems and employing KLR algebra isomorphisms.

Findings

Proves Morita equivalence for RoCK blocks and wreath products

Extends results to arbitrary fields and Hecke algebras

Uses isomorphism with cyclotomic 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.