Mohamad N. Nasser — Page 2

arXiv2021

On The Faithfulness of the extension of Lawrence-Krammer representation of the group of conjugating automorphisms C_3

Let $C_n$ be the group of conjugating automorphisms. We study the representation $ρ$ of $C_n$, an extension of Lawrence-Krammer representation of the braid group $B_n$, defined by Valerij G. Bardakov. As Bardakov proved that the representation $ρ$ is unfaithful for $n \geq 5$, the cases $n=3,4$ remain open. In our work, we make attempts towards the faithfulness of $ρ$ in the case $n=3$.

math.RT↗

arXiv2021

On the irreducibility of the extensions of Burau and Gassner representations

We study the $n$th degree representations $\hat{ρ_G}$ of $Cb_{n}$ and $\hat{ρ_B}$ 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 $\hat{ρ_G}$ is reducible and its $(n-1)$th degree composition factor $\hat{ϕ_G}$ is irreducible if and only if $t_i\neq 1$ for all $1\leq i \leq n$. Also we prove that $\hat{ρ_B}$ is reducible and its $(n-1)$th degree composition factor $\hat{ϕ_B}$ is irreducible if and only if $t\neq 1$. Moreover, for $n=3$, we prove that $\hat{ϕ_G}(t_1,t_2,t_3) \otimes \hat{ϕ_G}(m_1,m_2,m_3)$ 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 $\hat{ϕ_B}(t) \otimes \hat{ϕ_B}(m)$ is irreducible if and only if $t \neq m$.

math.GR↗