On extensions of the standard representation of the braid group to the singular braid group
Mohamad N. Nasser
TL;DR
This work extends the Tong–Yang–Ma standard representation $\rho_S$ of the braid group to two major group extensions, $SB_n$ and $VSB_n$, and provides a detailed classification of the resulting extended representations. It first characterizes all representations $\rho'_S: SB_n \to GL_n(\mathbb{Z}[t^{\pm1}])$ that extend $\rho_S$, giving explicit block-structured forms with parameters $a,c$ and showing how irreducibility depends on $t$ and $a+c$. It then proves irreducibility criteria, showing irreducibility for $t\neq1$ and, for $t=1$, irreducibility iff $a+c\neq1$, while establishing unfaithfulness for $n\ge3$ due to the known nonfaithfulness of $\rho_S$ and kernels containing the commutator subgroup of the pure braid group. Finally, it advances the study of $VSB_n$ by classifying all possible $\rho''_S: VSB_2 \to GL_2(\mathbb{Z}[t^{\pm1}])$ extending $\rho_S$ and $\rho'_S$, outlining a path toward similar classifications for $n\ge3$ and posing open questions on irreducibility and kernels for the $VSB_n$ extensions.
Abstract
For an integer $n \geq 2$, set $B_n$ to be the braid group on $n$ strands and $SB_n$ to be the singular braid group on $n$ strands. $SB_n$ is one of the important group extensions of $B_n$ that appeared in 1998. Our aim in this paper is to extend the well-known standard representation of $B_n$, namely $ρ_S:B_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, to $SB_n$, for all $n \geq 2$, and to investigate the characteristics of these extended representations as well. The first major finding in our paper is that we determine the form of all representations of $SB_n$, namely $ρ'_S: SB_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, that extend $ρ_S$, for all $n\geq 2$. The second major finding is that we find necessary and sufficient conditions for the irreduciblity of the representations of the form $ρ'_S$ of $SB_n$, for all $n\geq 2$. We prove that, for $t\neq 1$, the representations of the form $ρ'_S$ are irreducible and, for $t=1$, the representations of the form $ρ'_S$ are irreducible if and only if $a+c\neq 1.$ The third major result is that we consider the virtual singular braid group on $n$ strands, $VSB_n$, which is a group extension of both $B_n$ and $SB_n$, and we determine the form of all representations $ρ''_S: VSB_2 \to GL_2(\mathbb{Z}[t^{\pm 1}])$, that extend $ρ_S$ and $ρ'_S$; making a path toward finding the form of all representations $ρ''_S: VSB_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, that extend $ρ_S$ and $ρ'_S$, for all $n\geq 3$.