Parameters for which the Lawrence-Krammer representation is reducible

TL;DR

This paper investigates the conditions under which the Lawrence-Krammer representation of the braid group becomes reducible, identifying specific parameter values and their implications for the algebra's structure.

Contribution

It provides a complete characterization of parameter values leading to reducibility and constructs an explicit BMW algebra representation within the Lawrence-Krammer space.

Findings

The Lawrence-Krammer representation is generically irreducible.

Specialized parameters cause the representation to become reducible.

The BMW algebra is not semisimple at reducibility points.

Abstract

We show that the Lawrence-Krammer representation based on two parameters that was used by Bigelow and independently Krammer to show the linearity of the braid group is generically irreducible, but that when its parameters are specialized to some nonzero complex numbers, the representation is reducible. To do so, we construct a representation of the BMW algebra inside the Lawrence-Krammer space. As a representation of the braid group, this representation is equivalent to the Lawrence-Krammer representation, where the two parameters of the algebra are related to the parameters of the Lawrence-Krammer representation. We give all the complex values of the parameters for which the representation is reducible and describe the invariant subspaces in some cases. We show that for these values of the parameters and other values, the BMW algebra is not semisimple.