On the irreducibility of singular braid group representations
Mohamad N. Nasser
TL;DR
The article advances the representation theory of the singular braid group $SB_n$ by providing complete classifications of irreducible complex representations for small ranks: $SB_2 o GL_2(\\mathbb{C})$ and $SB_3 o GL_2(\\mathbb{C})$, along with selected $SB_3 o GL_3(\\mathbb{C})$ cases. It shows that $SB_2$ irreducibles are equivalent to a single form with $\\rho_3(\\sigma_1)=\\mathrm{diag}(w,w)$ and $\\rho_3(\\tau_1)=\\begin{pmatrix}a&b\\ c&d\\end{pmatrix}$ under $ad\\neq bc$, $bc\\neq 0$, $w\\neq 0$, and that $SB_3$ irreducibles arise as extensions of irreducible $B_3$ representations (Tuba–Wenzl) to $SB_3$, with explicit parameter relations for the images of the singular generators. The work further develops irreducibility criteria for homogeneous local representations of $SB_n$ (for all $n\\ge 3$), showing reducibility for certain families ($\\rho_1$, $\\rho_2$) and providing precise conditions under which $\\rho_3$ remains irreducible, thereby connecting local representation theory with classical braid-group representations. The results enhance the understanding of how singularities affect representation theory and establish concrete irreducibility criteria across different ranks and representation families.
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 $ρ: SB_2 \to GL_2(\mathbb{C})$. The second finding is the determination of the forms of all irreducible representations $ρ: SB_3 \to GL_2(\mathbb{C})$, together with the introduction of the forms of some irreducible representations $ρ: SB_3 \to GL_3(\mathbb{C})$. The third finding is the study of the irreducibility of the homogeneous local representations of $SB_n$ for all $n\geq 3$.