Rock blocks

TL;DR

This paper studies Rock blocks, special symmetric blocks in representation theory related to symmetric groups, Hecke algebras, and finite general linear groups, aiming to understand their structure and symmetries.

Contribution

It develops a structure theorem for Rock blocks, revealing their high symmetry and their derived equivalence to all blocks in the context of symmetric groups and related algebras.

Findings

Rock blocks are more symmetric than general blocks.

Every block is derived equivalent to a Rock block.

The paper establishes a structure theorem for Rock blocks.

Abstract

Consider representation theory associated to symmetric groups, or to Hecke algebras in type A, or to q-Schur algebras, or to finite general linear groups in non-describing characteristic. Rock blocks are certain combinatorially defined blocks appearing in such a representation theory, first observed by R. Rouquier. Rock blocks are much more symmetric than general blocks, and every block is derived equivalent to a Rock block. Motivated by a theorem of J. Chuang and R. Kessar in the case of symmetric group blocks of abelian defect, we pursue a structure theorem for these blocks.