On the irreducibility of the extensions of Burau and Gassner representations

Abstract

We study the $n$th degree representations $\widehat{{\rho }_G}$ of $C{b}_n$ and $\widehat{{\rho }_B}$ of $C_n$, defined by Valerij G. Bardakov, where $C{b}_n$ is the group of basis conjugating automorphisms and $C_n$ is the group of conjugating automorphisms. We prove that $\widehat{{\rho }_G}$ is reducible and its $(n-1)$th degree composition factor $\widehat{{\varphi }_G}$ is irreducible if and only if $t_i\ne 1$ for all $1\le i\le n$. Also we prove that $\widehat{{\rho }_B}$ is reducible and its $(n-1)$th degree composition factor $\widehat{{\varphi }_B}$ is irreducible if and only if $t\ne 1$. Moreover, for $n=3$, we prove that $\widehat{{\varphi }_G}(t_1,t_2,t_3)\otimes \widehat{{\varphi }_G}(m_1,m_2,m_3)$ is irreducible if and only if $(t_1,t_2,t_3)$ and $(m_1,m_2,m_3)$ are distinct vectors, and the representation $\widehat{{\varphi }_B}(t)\otimes \widehat{{\varphi }_B}(m)$ is irreducible if and only if $t\ne m$.