Tensor products of specializations of the Burau representation

The reduced Burau representation is a one-parameter representation of Bn, the braid group on n strings. Specializing the parameter to a nonzero complex number y gives a representation image which is either irreducible or has an irreducible composition factor image. We prove that the tensor product of an irreducible βn(y) or image with an irreducible βn(z) or image is irreducible unless y=z±1.