Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type $A_{n-1}$

TL;DR

This paper investigates the irreducibility of the Lawrence-Krammer representation of the BMW algebra of type A_{n-1}, identifying parameter values that lead to reducibility and discussing implications for algebra semisimplicity.

Contribution

It characterizes all parameter specializations causing reducibility of the Lawrence-Krammer representation and explores consequences for the BMW algebra's semisimplicity.

Findings

The representation is generically irreducible.

Certain parameter values cause reducibility.

Results on semisimplicity of BMW algebra.

Abstract

It is known that the Lawrence-Krammer representation of the Artin group of type $A_{n-1}$ based on the two parameters $t$ and $q$ that was used by Krammer and independently by Bigelow to show the linearity of the braid group on $n$ strands is generically irreducible. Here, we recover this result and show further that for some complex specializations of the parameters the representation is reducible. We give all the values of the parameters for which the representation is reducible as well as the dimensions of the invariant subspaces. We deduce some results of semisimplicity of the Birman-Murakami-Wenzl algebra of type $A_{n-1}$.