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

Valerij G. Bardakov and P. Bellingeri introduced a new linear representation \(\bar{\rho }_F\) ${\overline{\rho }}_F$ of degree \(n+1\) $n+1$ of the braid group \(B_n\) $B_n$ . We study the irreducibility of this representation. We prove that \(\bar{\rho }_F\) ${\overline{\rho }}_F$ is reducible to the degree \(n-1\) $n-1$ . Moreover, we give necessary and sufficient conditions for the irreducibility of the complex specialization of its \(n-1\) $n-1$ degree composition factor \(\bar{\phi }_F\) ${\overline{\varphi }}_F$ .