On the irreducibility of singular braid group representations

TL;DR

This paper investigates the irreducibility of singular braid group representations, classifying irreducible representations for small cases and exploring conditions for irreducibility in general.

Contribution

It provides a complete classification of irreducible representations of SB_2 and SB_3 into GL_2 and GL_3, and studies irreducibility of local representations for all n ≥ 3.

Findings

Classified all irreducible representations of SB_2 in GL_2(C)

Determined all irreducible representations of SB_3 in GL_2(C) and some in GL_3(C)

Analyzed irreducibility of homogeneous local representations for all n ≥ 3

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 $n\geq 3$.