REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE D_n

Abstract

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.