RoCK blocks for affine categorical representations

TL;DR

This paper classifies Scopes equivalence classes of blocks in affine categorifications using hyperplane arrangements, introduces RoCK blocks, and provides computational tools for their identification in Ariki-Koike algebras.

Contribution

It introduces RoCK blocks for affine categorical representations, linking them to hyperplane arrangements and providing Sage code for their analysis.

Findings

Classification of Scopes equivalence classes via hyperplane chambers

Identification of RoCK blocks as largest equivalence classes

Provision of Sage code for testing and finding RoCK blocks

Abstract

Given a categorical action of a Lie algebra, a celebrated theorem of Chuang and Rouquier proves that the blocks corresponding to weight spaces in the same orbit of the Weyl group are derived equivalent, proving an even more celebrated conjecture of Brou\'e for the case of the symmetric group. In many cases, these derived equivalences are $t$-exact, and thus induce equivalences of abelian categories between different blocks. We call two such blocks ``Scopes equivalent.'' In this paper, we describe how Scopes equivalence classes for any affine categorification can be classified by the chambers of a finite hyperplane arrangement, which can be found through simple Lie theoretic calculations. We pay special attention to the largest equivalence classes, which we call RoCK, and show how this matches with recent work of Lyle on Rouquier blocks for Ariki-Koike algebras. We also provide Sage code that tests whether blocks are RoCK and finds RoCK blocks for Ariki-Koike algebras.