Data

1. Title: Semisimplicity of Hecke and (walled) Brauer algebras
2. Authors: Henning Haahr Andersen, Catharina Stroppel and Daniel Tubbenhauer
3. Status: J. Aust. Math. Soc. 103 (2017), no. 1, 1-44. Last update: Sat, 10 Sep 2016 13:04:44 GMT
5. ArXiv version = 0.99 published version
6. LaTex Beamer presentation: Slides1, Slides2

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.

A few extra words

In this paper we give a semisimplicity criterion for the algebra $\mathrm{End}_{\textbf{U}_q}(T)$ which only relies on the combinatorics of the root and weight data associated to $\mathfrak{g}$. The crucial observation we use here is that $\mathrm{End}_{\textbf{U}_q}(T)$ is semisimple iff all Weyl factors of $T$ are simple $\textbf{U}_q$-modules -- a question which can be checked using (versions of) Jantzen's sum formula. Here $\textbf{U}_q=\textbf{U}_q(\mathfrak{g})$ is the $q$-deformed enveloping algebra of $\mathfrak{g}$ (any reductive Lie algebra)) over $\mathbb{K}$ and $T$ is a $\textbf{U}_q$-tilting module (note that in the semisimple case, e.g. the classical case $\mathbb{K}=\mathbb{C}$ and $q=1$, all $\textbf{U}_q$-modules are $\textbf{U}_q$-tilting modules - so the whole classical story is included).
We apply our methods to four explicit examples: the Hecke algebras of types $\textbf{A}$ and $\textbf{B}$, the walled Brauer algebras and the Brauer algebras. For all of these we obtain full semisimplicity criteria by using the corresponding combinatorics of roots and weights: our approach has the advantage that it provides a more general method to deduce semisimplicity criteria and the calculations are always the same (mutatis mutandis, depending on the associated root and weight data). Hence, our approach unites the known semisimplicity criteria of these algebras in our more general framework.
In fact, we could “quantize” our results on semisimplicity criteria and obtain them for quantized walled Brauer algebras and Birman-Murakami-Wenzl algebras as well (for brevity, we decided not to). Even better: our methods to deduce semisimplicity criteria work more generally, e.g. for $\boldsymbol{\mathcal{O}}$ and its tilting theory.
As a by-product we also obtain diagrammatic versions of some of the kernels of the Schur-Weyl(-Brauer) actions. For example, such a kernel element in the walled Brauer case looks like: