On the irreducibility of singular braid group representations

Abstract

In this article, we study the irreducibility of representations of the singular braid group on $n$ strands, namely $SB_n$. Our first finding is the determination of the forms of all irreducible representations $\rho : SB_2 \to GL_2(\mathbb{C})$. The second finding is the determination of the forms of all irreducible representations $\rho : SB_3 \to GL_2(\mathbb{C})$, together with the introduction of the forms of some irreducible representations $\rho : SB_3 \to GL_3(\mathbb{C})$. The third finding is the study of the irreducibility of the homogeneous local representations of $SB_n$ for all $n\geq 3$.