On the irreducibility of linear representations of the pure braid group

Following up on our result in [1], we find a milder sufficient condition for the tensor product of specializations of the reduced Gassner representation of the pure braid group to be irreducible. We prove that $G_n(x_1, \ldots, x_n) \otimes G_n(y_1, \ldots, y_n) : P_n \to GL(\mathbb{C}^{n-1} \otimes \mathbb{C}^{n-1})$ is irreducible if $x_i \neq \pm y_i $ and $x_j \neq \pm {{y_j}^{-1}} $ for some $i$ and $j$.