On irreducible representations of quandles

TL;DR

This paper studies the structure of irreducible representations of finite quandles over complex numbers, relating them to characters, automorphism groups, and projective representations, with applications to conjugacy quandles of specific groups.

Contribution

It provides a framework to construct irreducible quandle representations from characters and group representations, especially for quandles with trivial Schur multipliers.

Findings

Irreducible representations of quandles can be built from characters and automorphism group representations.

Results apply to conjugacy quandles of dihedral and quaternion groups.

Constructs relate quandle representations to projective group representations and group quotients.

Abstract

We consider irreducible representations of finite quandles over $\mathbb{C}$. For $Q$ a finite quandle whose inner automorphism group $Inn(Q)$ have trivial Schur multipliers, we prove that the irreducible representations of $Q$ can be constructed out of what we call characters of $Q$ and irreducible linear represenations of the group $Inn(Q)$. For $G$ a finite groiup having trivial Schur multiplier or being a Schur cover of another group, we show that the irreducible representations of the conjugacy quandle $Conj(G)$ can be constructed out of characters of $Conj(G)$ and irreducible linear representations of the group $G$. In both cases, the finite unitary irreducible representations can be determined from the results. For instance, these results allow to solve the problem of constucting irreducible represenations of the conjugacy quandles of dihedral groups and generalised quaternion groups. In general, we relate the irreducible representations of a finite quandle $Q$ to irreducible projective representations of $Inn(Q)$ and prove that the irreducible representations of $Q$ can be in theory constructed out of characters of $Q$ and irreducible representations of a finite quotient of the enveloping group $G(Q)$. The quotient is a stem extensions of $Inn(Q)$ with nucleus a finite subgroup of the center of $G(Q)$. This allows, using a result from the litterature, to show that the irreducible quandle representations of $Conj(S_n)$ ($S_n$ the symmetric group) can be constructed out of characters of the corresponding quandle and irreducible linear group representations of the symmetric group.