Complex specializations of the reduced Gassner representation of the pure braid group

We will give a necessary and sufficient condition for the specialization of the reduced Gassner representation $G_{n}(z): P_{n} \to GL_{n-1}(\mathbb {C})$ to be irreducible. It will be shown that for $z=(z_{1},\ldots ,z_{n})\in (\mathbb {C}^{*})^{n}$, $G_{n}(z)$ is irreducible if and only if $z_{1}\ldots z_{n} \neq 1$.