Specht modules and semisimplicity criteria for Brauer and Birman--Murakami--Wenzl Algebras

TL;DR

This paper constructs cellular bases for Birman--Murakami--Wenzl algebras, generalizes Specht module theory, and provides explicit semisimplicity criteria based on parameters q and r.

Contribution

It introduces a new method to build cellular bases for B--M--W algebras and derives semisimplicity conditions using an abelian subalgebra acting triangularly.

Findings

Cellular bases for B--M--W algebras constructed

Explicit semisimplicity criteria in terms of q and r

Generalization of Specht module theory results

Abstract

A construction of bases for cell modules of the Birman--Murakami--Wenzl (or B--M--W) algebra $B_n(q,r)$ by lifting bases for cell modules of $B_{n-1}(q,r)$ is given. By iterating this procedure, we produce cellular bases for B--M--W algebras on which a large abelian subalgebra, generated by elements which generalise the Jucys--Murphy elements from the representation theory of the Iwahori--Hecke algebra of the symmetric group, acts triangularly. The triangular action of this abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters $q$ and $r$, for B--M--W algebras to be semisimple. The aforementioned constructions provide generalisations, to the algebras under consideration here, of certain results from the Specht module theory of the Iwahori--Hecke algebra of the symmetric group.