Reducibility of the Cohen-Wales representation of the Artin group of type $D_n$

Abstract

Using knot theory, we construct a linear representation of the CGW algebra of type $D_n$. This representation has degree $n^2-n$, the number of positive roots of a root system of type $D_n$. We show that the representation is generically irreducible, but that when the parameters of the algebra are related in a certain way, it becomes reducible. As a representation of the Artin group of type $D_n$, this representation is equivalent to the faithful linear representation of Cohen-Wales. We give a reducibility criterion for this representation as well as a conjecture on the semisimplicity of the CGW algebra of type $D_n$. Our proof is computer-assisted using Mathematica.

55 pages, 22 figures

References in corpus (2)

• Specht modules and semisimplicity criteria for Brauer and Birman--Murakami--Wenzl Algebras
• Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type $A_{n-1}$, PhD thesis California Institute of Technology 2008

Cited by in corpus (2)

• Classification of the invariant subspaces of the Cohen-Wales representation of the Artin group of type $D_n$
• Tangles of type $E_n$ and a reducibility criterion for the Cohen-Wales representation of the Artin group of type $E_6$