On the irreducibility of the extensions of Burau and Gassner representations

TL;DR

This paper investigates the reducibility of certain automorphism group representations, establishing conditions under which their composition factors are irreducible, with specific results for the cases when n=3.

Contribution

It proves the reducibility of the representations of $Cb_{n}$ and $C_{n}$ and characterizes when their composition factors are irreducible, including the special case n=3.

Findings

$ ho_G$ is reducible; $ ext{phi}_G$ is irreducible iff all $t_i eq 1$.

$ ho_B$ is reducible; $ ext{phi}_B$ is irreducible iff $t eq 1$.

For $n=3$, tensor products are irreducible under specific conditions on parameters.

Abstract

We study the $n$th degree representations $\hat{\rho_G}$ of $Cb_{n}$ and $\hat{\rho_B}$ of $C_{n}$, defined by Valerij G. Bardakov, where $Cb_{n}$ is the group of basis conjugating automorphisms and $C_n$ is the group of conjugating automorphisms. We prove that $\hat{\rho_G}$ is reducible and its $(n-1)$th degree composition factor $\hat{\phi_G}$ is irreducible if and only if $t_i\neq 1$ for all $1\leq i \leq n$. Also we prove that $\hat{\rho_B}$ is reducible and its $(n-1)$th degree composition factor $\hat{\phi_B}$ is irreducible if and only if $t\neq 1$. Moreover, for $n=3$, we prove that $\hat{\phi_G}(t_1,t_2,t_3) \otimes \hat{\phi_G}(m_1,m_2,m_3)$ 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 $\hat{\phi_B}(t) \otimes \hat{\phi_B}(m)$ is irreducible if and only if $t \neq m$.