On the irreducibility of singular braid group representations

In this article, we study the irreducibility of representations of the singular braid group on $n$ strands, namely $S{B}_n$. Our first finding is the determination of the forms of all irreducible representations $\rho :S{B}_2\to G{L}_2(\mathbb{C})$. The second finding is the determination of the forms of all irreducible representations $\rho :S{B}_3\to G{L}_2(\mathbb{C})$, together with the introduction of the forms of some irreducible representations $\rho :S{B}_3\to G{L}_3(\mathbb{C})$. The third finding is the study of the irreducibility of the homogeneous local representations of $S{B}_n$ for all $n\ge 3$.