Vertices for Iwahori-Hecke algebras and the Dipper-Du conjecture

TL;DR

This paper establishes a classification of vertices of blocks of Iwahori-Hecke algebras of symmetric groups in any characteristic, and uses it to resolve the Dipper-Du conjecture concerning the structure of vertices of indecomposable modules.

Contribution

It introduces a Green and Brauer correspondence framework for Hecke algebra bimodules, computes vertices of Specht modules, and classifies block vertices to prove the Dipper-Du conjecture.

Findings

Classification of vertices of blocks of $ ext{Iwahori-Hecke}$ algebras.

Resolution of the Dipper-Du conjecture on vertices of indecomposable modules.

Development of Green and Brauer correspondences for Hecke algebra bimodules.

Abstract

Let $\mathscr{H}_n$ denote the Iwahori-Hecke algebra corresponding to the symmetric group $\mathfrak{S}_n$. We set up a Green correspondence for bimodules of these Hecke algebras, and a Brauer correspondence between their blocks. We examine Specht modules for $\mathscr{H}_n$ and compute the vertices of certain Specht modules, before using this to give a classification of the vertices of blocks of $\mathscr{H}_n$ in any characteristic. Finally, we apply this classification to resolve the Dipper-Du conjecture about the structure of vertices of indecomposable $\mathscr{H}_n$-modules.