On the irreducibility of singular braid group representations

Abstract

In this article, we investigate the irreducibility of representations of the singular braid group on $n$ strands, denoted by $SB_n$. We first show that all representations $\rho: SB_2 \longrightarrow \mathrm{GL}_2(\mathbb{C})$ are reducible. We then determine all irreducible representations $\rho: SB_3 \longrightarrow \mathrm{GL}_2(\mathbb{C})$ and construct several families of irreducible representations $\rho: SB_3 \longrightarrow \mathrm{GL}_3(\mathbb{C})$. Finally, we study the irreducibility of homogeneous local representations of $SB_n$ for all $n \geq 3$.