On irreducible representations of conjugacy quandles

Abstract

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 $G$ ($\alpha (g,h)=\alpha (h,g)$ 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.

Error in the first version (v1)