The interplay between linear representations of the braid group. (Q925446)

Summary: We consider Wada's representation as a twisted version of the standard action of the braid group, \(B_n\), on the free group with \(n\) generators. Constructing a free group, \(G_{nm}\), of rank \(nm\), we compose Cohen's map \(B_n\to B_{nm}\) and the embedding \(B_{nm}\to\Aut(G_{nm})\) 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 \(t^k\) 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.