WADA'S REPRESENTATIONS OF THE PURE BRAID GROUP

We consider the Magnus representation of the image of the pure braid group under the generalizations of the standard Artin representation, discovered by M. Wada. We will give a necessary and sufficient condition for the specialization of the reduced Wada's representation G_n(z) : P_n → GL_n-1(ℂ) to be irreducible. It will be shown that for z = (z₁,…,z_n) ∈ (ℂ*)ⁿ, G_n(z) is irreducible if and only if z₁^k⋯z_n^k ≠ 1. This is a generalization of our previous result concerning the irreducibility of the complex specialization of the reduced Gassner representation of P_n.