On the representation theory of cyclic and dihedral quandles

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.