Tensor Products of the Gassner Representation of The Pure Braid Group†

ABSTRACT

The reduced Gassner representation is a multi-parameter representation of Pn, the pure braid group on n strings. Specializing the parameters t1, t2,...,tn to nonzero complex numbers x1,x2,...,xn gives a representation Gn (x1,...,xn): Pn → GL (Cn-1) which is irreducible if and only if x1...xn ≠ 1.We find a sufficient condition that guarantees that the tensor product of an irreducible Gn (x1,...,xn) with an irreducible Gn (y1, ..., yn) is irreducible. We fall short of finding a necessary and sufficient condition for irreducibility of the tensor product. Our work is a continuation of a previous one regarding the tensor product of complex specializations of the Burau representation of the braid group.

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