Parameters for which the Lawrence–Krammer representation is reducible

Abstract

We show that the representation, introduced by Lawrence and Krammer to show the linearity of the braid group, is generically irreducible. However, for some values of its two parameters when these are specialized to complex numbers, it becomes reducible. We construct a representation of degree n(n−1)/2 of the BMW algebra of type A_(n−1). As a representation of the braid group on n strands, it is equivalent to the Lawrence–Krammer representation where the two parameters of the BMW algebra are related to those appearing in the Lawrence–Krammer representation. We give the values of the parameters for which the representation is reducible and give the proper invariant subspaces in some cases. We use this representation to show that for these special values of the parameters, the BMW algebra of type A_(n−1) is not semisimple.

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