Intersections of blocks of cyclotomic Hecke algebras

TL;DR

This paper proves a conjecture about the intersections of blocks in cyclotomic Hecke algebras for most exceptional groups and explores broader generalizations to other algebraic structures.

Contribution

It confirms Trinh and Xue's conjecture for all exceptional types except E8 and extends the ideas to Suzuki, Ree, Coxeter, and spetsial complex reflection groups.

Findings

Proved the conjecture for all exceptional groups except E8.

Proposed and confirmed generalizations to other algebraic groups.

Extended understanding of block intersections in complex reflection groups.

Abstract

Trinh and Xue have proposed a startling conjecture on intersections of blocks of cyclotomic Hecke algebras occurring in modular representation theory of finite reductive groups. We prove this conjecture for all exceptional type groups apart from $E_8$. We also propose several generalisations, to Suzuki and Ree groups, to non-rational Coxeter groups and even more generally to spetsial complex reflection groups, and confirm these in various cases.