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

Abstract

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