Necessary and sufficient conditions for the irreducibility of a linear representation of the braid group \(B_n\) (Q6611206)

Let \(B_n\) be the braid group on \(n\) strings and \(F_m\) be the free group of rank \(m\). There are several representations of \(B_n\) into the automorphism group of \(F_m\). The most popular among them is the Artin representation \(B_n \to \Aut(F_n)\), and the extended Artin representation \(B_n \to \Aut(F_{n+1})\). Another one is the Fenn-Rolfsen-Zhu representation \(B_n \to \Aut(F_{n+1})\), and the reduced Fenn-Rolfsen-Zhu representation \(B_n \to \Aut(F_{n+1})\). \textit{V. G. Bardakov} and \textit{P. Bellingeri} [Contemp. Math. 670, 285--298 (2016; Zbl 1368.20040)] explored relationships between several different representations of \(B_n\) as automorphisms of free groups. In particular, they proved that the extended Artin representation is conjugated to Fenn-Rolfsen-Zhu representation, and that reduced Fenn-Rolfsen-Zhu representation is conjugated to Artin representation. Also, they constructed a new linear Burau-like representation \(\bar{\rho}_F \colon B_n \to GL_{n+1}(\mathbb{Z}[t^{\pm 1}])\). This representation is induced by the representations of Fenn-Rolfsen-Zhu and Perron-Vannier.\N\NThe aim of this paper is to study the irreducibility of the representation \(\bar{\rho}_F\). It is proven that \(\bar{\rho}_F\) is reducible and the complex specialization of its \(n - 1\) composition factor, \(\bar{\phi}_F \colon B_n \to GL_{n-1}(\mathbb{C})\), is irreducible if and only if \(z\) is not a root of the polynomial \(g_n(t) = t^{n-1} + 2 t^{n-2} + 3 t^{n-3} + \ldots + (n - 1)t + n\).