Conjugation in Brauer algebras and applications to character theory

TL;DR

This paper develops a conjugation theory within Brauer algebras, enabling the definition of modular characters and extending the classical character theory to non-semisimple cases, thus broadening the algebra's representation understanding.

Contribution

It introduces and fully characterizes three types of conjugation in the Brauer algebra monoid, leading to new modular character definitions and extensions of existing character theory.

Findings

Defined three types of conjugation in the Brauer algebra monoid

Established properties of modular characters for Brauer algebras

Extended classical character theory to non-semisimple Brauer algebras

Abstract

The Brauer algebra has a basis of diagrams and these generate a monoid $H$ consisting of scalar multiples of diagrams. Following a recent paper by Kudryavtseva and Mazorchuk, we define and completely determine three types of conjugation in $H$. We are thus able to define modular characters for Brauer algebras which share many of the properties of Brauer characters defined for finite groups over a field of prime characteristic. Furthermore, we reformulate and extend the theory of ordinary characters for Brauer algebras as introduced by Ram to the case when the Brauer algebra is not semisimple.