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 $S{B}_n$. We first show that all representations $\rho :S{B}_2⟶{\mathrm{GL}}_2(\mathbb{C})$ are reducible. We then determine all irreducible representations $\rho :S{B}_3⟶{\mathrm{GL}}_2(\mathbb{C})$ and construct several families of irreducible representations $\rho :S{B}_3⟶{\mathrm{GL}}_3(\mathbb{C})$. Finally, we study the irreducibility of homogeneous local representations of $S{B}_n$ for all $n\ge 3$.