Semisimplicity of Hecke and (walled) Brauer algebras

TL;DR

This paper develops a unified approach using Jantzen's sum formula to establish semisimplicity criteria for various endomorphism algebras, including Hecke, Brauer, and walled Brauer algebras, across different fields and parameters.

Contribution

It introduces a general method for proving semisimplicity of endomorphism algebras of tilting modules, recovering known criteria for multiple algebra types.

Findings

Semisimplicity criteria for Hecke algebras of types A and B.

Semisimplicity criteria for walled Brauer and Brauer algebras.

Unified approach applicable over any field and parameter.

Abstract

We show how to use Jantzen's sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\textbf{U}_q$-tilting modules (for any field $\mathbb{K}$ and any parameter $q\in\mathbb{K}-\{0,-1\}$). As an application, we recover the semisimplicity criteria for the Hecke algebras of types $\textbf{A}$ and $\textbf{B}$, the walled Brauer algebras and the Brauer algebras from our more general approach.