On irreducible representations of conjugacy quandles

TL;DR

This paper characterizes all irreducible quandle representations of conjugacy quandles of finite groups over complex numbers, linking them to symmetric 2-cocycles and group cohomology.

Contribution

It provides a complete classification of irreducible conjugacy quandle representations using group cohomology conditions, especially for groups with trivial Bogomolov multiplier.

Findings

All irreducible quandle representations are products of group representations and quandle characters under certain conditions.

For groups with trivial symmetric 2-cocycle cohomology, the enveloping group injects into a product involving G and a free abelian group.

If G is perfect, the enveloping group of the conjugacy quandle is isomorphic to G times a free abelian group.

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.