RoCK blocks for double covers of symmetric groups and quiver Hecke superalgebras

TL;DR

This paper introduces RoCK blocks for double covers of symmetric groups, proves their Morita equivalence to local blocks, and confirms Broué's conjecture for these blocks using categorification and superalgebra techniques.

Contribution

It extends the theory of RoCK blocks to double covers and quiver Hecke superalgebras, establishing Morita equivalences and proving Broué's conjecture in this context.

Findings

RoCK blocks of double covers are Morita equivalent to local blocks.

Broué's abelian defect group conjecture is proved for these RoCK blocks.

Classification of irreducible representations via cuspidal systems.

Abstract

We define and study RoCK blocks for double covers of symmetric groups. We prove that RoCK blocks of double covers are Morita equivalent to standard `local' blocks. The analogous result for blocks of symmetric groups, a theorem of Chuang and Kessar, was an important step in Chuang and Rouquier ultimately proving Brou\'e's abelian defect group conjecture for symmetric groups. Indeed we prove Brou\'e's conjecture for the RoCK blocks defined in this article. Our methods involve translation into the quiver Hecke superalgebras setting using the Kang-Kashiwara-Tsuchioka isomorphism and categorification methods of Kang-Kashiwara-Oh. As a consequence we construct Morita equivalences between more general objects than blocks of finite groups. In particular, our results extend to certain blocks of level one cyclotomic Hecke-Clifford superalgebras. We also study imaginary cuspidal categories of quiver Hecke superalgebras and classify irreducible representations of quiver Hecke superalgebras in terms of cuspidal systems.