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( n 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.

Abstract. This work is directed towards the open question of the faithfulness of the reduced Gassner representation of the pure braid group, Pn(n > 3). Long and Paton proved that if a Burau matrix M has ones on the diagonal and zeros below the diagonal then M is the identity matrix. In this paper, a generalization of Long and Paton's result will be proved. Our main theorem is that if the trace of the image of an element of Pn under the reduced Gassner representation is n − 1, then this element lies in the kernel of this representation. Then, as a corollary, we prove that an analogue of the main theorem holds true for the Burau representation of the braid group.

Abstract. We will give a necessary and sufficient condition for the specialization of the reduced Gassner representation Gn(z) : Pn → GL n−1 (C) to be irreducible. It will be shown that for z = (z 1 , . . . , zn) ∈ (C * ) n , Gn(z) is irreducible if and only if z 1 . . . zn = 1.

We show that the reduced Gassner representation restricted to a normal free subgroup U n of the pure braid group, P n , is essentially the Magnus representation of U n = U 0 0 n provided that the indeterminate y n , used in defining this representation, is set to be one. Then, as a corollary, Lipschutz's result follows easily that is, in general, the kernel of the reduced Gassner representation restricted to U n is a subgroup of U 0 0 n .

The reduced Burau representation is a one-parameter representation of B n , the braid group on n strings. Specializing the parameter to a nonzero complex number y gives a representation n (y) : B n → GL(C n−1 ) which is either irreducible or has an irreducible composition factorˆ n (y) : B n → GL(C n−2 ). We prove that the tensor product of an irreducible n (y) orˆ n (y) with an irreducible n (z) orˆ n (z) is irreducible unless y = z ±1 .