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 βn(y):Bn→GL(Cn-1) which is either irreducible or has an irreducible composition factor β^n(y):Bn→GL(Cn-2). We prove that the tensor product of an irreducible βn(y) or β^n(y) with an irreducible βn(z) or β^n(z) is irreducible unless y=z±1.