On extensions of the standard representation of the braid group to the singular braid group

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 $\rho_S:B_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, 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 $\rho'_S: SB_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, that extend $\rho_S$, 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 $\rho'_S$ of $SB_n$, for all $n\geq 2$. We prove that, for $t\neq 1$, the representations of the form $\rho'_S$ are irreducible and, for $t=1$, the representations of the form $\rho'_S$ 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 $\rho''_S: VSB_2 \to GL_2(\mathbb{Z}[t^{\pm 1}])$, that extend $\rho_S$ and $\rho'_S$; making a path toward finding the form of all representations $\rho''_S: VSB_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, that extend $\rho_S$ and $\rho'_S$, for all $n\geq 3$.