The Interplay between Linear Representations of the Braid Group

Abstract

We consider Wada's representation as a twisted version of the standard action of the braid group, Bn, on the free group with n generators. Constructing a free group, Gnm, of rank nm, we compose Cohen's map Bn→Bnm and the embedding Bnm→Aut(Gnm) via Wada's map. We prove that the composition factors of the obtained representation are one copy of Burau representation and m−1 copies of the standard representation after changing the parameter t to tk in the definitions of the Burau and standard representations. This is a generalization of our previous result concerning the standard Artin representation of the braid group.