On the representation theory of cyclic and dihedral quandles

TL;DR

This paper explores the representation theory of cyclic and dihedral quandles, introducing regular representations, classifying them, and comparing their properties to finite group representations, including the failure of Maschke's theorem.

Contribution

It introduces the notion of regular quandle representations, classifies dihedral and cyclic quandle representations, and highlights differences from group representation theory.

Findings

Complete classification of dihedral quandle representations

Necessary conditions for irreducibility of cyclic quandle representations

Maschke's theorem does not hold for quandle representations

Abstract

Quandle representations are homomorphisms from a quandle to the group of invertible matrices on some vector space taken with the conjugation operation. We study certain families of quandle representations. More specifically, we introduce the notion of regular representation for quandles, investigating in detail the regular representations of dihedral quandles and \emph{completely classifying} them. Then, we study representations of cyclic quandles, giving some necessary conditions for irreducibility and providing a complete classification under some restrictions. Moreover, we provide various counterexamples to constructions that hold for group representations, and show to what extent such theory has the same properties of the representation theory of finite groups. In particular, we show that Maschke's theorem does not hold for quandle representations.