Narrow Results

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.

We consider the Magnus representation of the image of the braid group under a representation discovered by Wada. We first investigate the question about the reducibility of this representation and show that it is of Burau type. Next, we show that the images of the generators of the braid group under that representation are unitary relative to a hermitian matrix. This is similar to the well known result by Squier that asserts that the Burau representation of the braid group is unitary.

In this note, we show that a homomorphism from the braid group B_n into an arbitrary group is injective if and only if its image can be left ordered in a particular way. It follows from this criterion that the image of B_n (n ≥ 2) under any homomorphism is bi-orderable if and only if it is infinite cyclic. As a second application, this criterion also provides a new proof that the Artin representation of B_n into Aut(F_n) is injective.