On irreducible representations of conjugacy quandles
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914530237349888 |
|---|---|
| author | Maassarani, Mohamad |
| author_facet | Maassarani, Mohamad |
| contents | For $G$ a finite group, one way to construct irreducible quandle representations over $\mathbb{C}$ of the conjugacy quandle $Conj(G)$ is by taking the product of an irreducible linear group representation of $G$ by what we call a quandle character of $Conj(G)$ (a quandle morphism into $\mathbb{C}^\times$ ). We show that these are all the irreducible quandle representations of $Conj(G)$ over $\mathbb{C}$ if and only if all the symmetric $2$-cocyles over $G$ ($α(g,h)=α(h,g)$ for all $g,h$) with values in $\mathbb{C}^\times$ are coboundaries. For instance, this is the case of groups with trivial Bogomolov multiplier. We apply this to study the enveloping group of $Conj(G)$. If $G$ finite satisfies the previous condition on symmetric $2$-cocycles, we obtain that the enveloping group of $Conj(G)$ injects into $G\times \mathbb{Z}^{c_G}$ where $c_G$ is the number of the conjugacy classes of $G$. If moreover $G$ is perfect the injection is an isomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22078 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On irreducible representations of conjugacy quandles Maassarani, Mohamad Representation Theory Group Theory For $G$ a finite group, one way to construct irreducible quandle representations over $\mathbb{C}$ of the conjugacy quandle $Conj(G)$ is by taking the product of an irreducible linear group representation of $G$ by what we call a quandle character of $Conj(G)$ (a quandle morphism into $\mathbb{C}^\times$ ). We show that these are all the irreducible quandle representations of $Conj(G)$ over $\mathbb{C}$ if and only if all the symmetric $2$-cocyles over $G$ ($α(g,h)=α(h,g)$ for all $g,h$) with values in $\mathbb{C}^\times$ are coboundaries. For instance, this is the case of groups with trivial Bogomolov multiplier. We apply this to study the enveloping group of $Conj(G)$. If $G$ finite satisfies the previous condition on symmetric $2$-cocycles, we obtain that the enveloping group of $Conj(G)$ injects into $G\times \mathbb{Z}^{c_G}$ where $c_G$ is the number of the conjugacy classes of $G$. If moreover $G$ is perfect the injection is an isomorphism. |
| title | On irreducible representations of conjugacy quandles |
| topic | Representation Theory Group Theory |
| url | https://arxiv.org/abs/2604.22078 |