On extensions of the standard representation of the braid group to the singular braid group
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910910540414976 |
|---|---|
| author | Nasser, Mohamad N. |
| author_facet | Nasser, Mohamad N. |
| contents | For an integer $n \geq 2$, set $B_n$ to be the braid group on $n$ strands and $SB_n$ to be the singular braid group on $n$ strands. $SB_n$ is one of the important group extensions of $B_n$ that appeared in 1998. Our aim in this paper is to extend the well-known standard representation of $B_n$, namely $ρ_S:B_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, to $SB_n$, for all $n \geq 2$, and to investigate the characteristics of these extended representations as well. The first major finding in our paper is that we determine the form of all representations of $SB_n$, namely $ρ'_S: SB_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, that extend $ρ_S$, for all $n\geq 2$. The second major finding is that we find necessary and sufficient conditions for the irreduciblity of the representations of the form $ρ'_S$ of $SB_n$, for all $n\geq 2$. We prove that, for $t\neq 1$, the representations of the form $ρ'_S$ are irreducible and, for $t=1$, the representations of the form $ρ'_S$ are irreducible if and only if $a+c\neq 1.$ The third major result is that we consider the virtual singular braid group on $n$ strands, $VSB_n$, which is a group extension of both $B_n$ and $SB_n$, and we determine the form of all representations $ρ''_S: VSB_2 \to GL_2(\mathbb{Z}[t^{\pm 1}])$, that extend $ρ_S$ and $ρ'_S$; making a path toward finding the form of all representations $ρ''_S: VSB_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, that extend $ρ_S$ and $ρ'_S$, for all $n\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09256 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On extensions of the standard representation of the braid group to the singular braid group Nasser, Mohamad N. Representation Theory For an integer $n \geq 2$, set $B_n$ to be the braid group on $n$ strands and $SB_n$ to be the singular braid group on $n$ strands. $SB_n$ is one of the important group extensions of $B_n$ that appeared in 1998. Our aim in this paper is to extend the well-known standard representation of $B_n$, namely $ρ_S:B_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, to $SB_n$, for all $n \geq 2$, and to investigate the characteristics of these extended representations as well. The first major finding in our paper is that we determine the form of all representations of $SB_n$, namely $ρ'_S: SB_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, that extend $ρ_S$, for all $n\geq 2$. The second major finding is that we find necessary and sufficient conditions for the irreduciblity of the representations of the form $ρ'_S$ of $SB_n$, for all $n\geq 2$. We prove that, for $t\neq 1$, the representations of the form $ρ'_S$ are irreducible and, for $t=1$, the representations of the form $ρ'_S$ are irreducible if and only if $a+c\neq 1.$ The third major result is that we consider the virtual singular braid group on $n$ strands, $VSB_n$, which is a group extension of both $B_n$ and $SB_n$, and we determine the form of all representations $ρ''_S: VSB_2 \to GL_2(\mathbb{Z}[t^{\pm 1}])$, that extend $ρ_S$ and $ρ'_S$; making a path toward finding the form of all representations $ρ''_S: VSB_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, that extend $ρ_S$ and $ρ'_S$, for all $n\geq 3$. |
| title | On extensions of the standard representation of the braid group to the singular braid group |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2504.09256 |