On quandle representations

Abstract

A unitary finite dimensional quandle representation is decomposable into a direct sum of irreducible represenations. Not all quandle representations satisfy this property. We prove that a finite dimensional quandle represenation $ρ:Q \to GL(V) $ of a finite quandle $Q$ over $\mathbb{C}$ is decomposable into a direct sum of irreducibles if and only if every element in the image of $ρ$ is diagonlizable. We show that an irreducible representation $ρ:Q \to GL(V)$ of a finite quandle over $\mathbb{C}$ is unitary for some inner product if and only if every element of the image of $ρ$ has determinant of modulus $1$. It follows that any irreducible representation of a finite quandle $Q$ over $\mathbb{C}$ can be twisted by a quandle character to obtain a unitary irreducible representation. We also prove that the enveloping group $G(Q)$, of a finite quandle $Q$, admit a faithfull finite dimensional unitary representation over $\mathbb{C}$ and that the irreducible representations of a finite quandle $Q$ over $\mathbb{C}$ are $1$-dimensional if and only if $G(Q)$ is abelian. Finaly, we determine the irreducible representations over $\mathbb{C}$ of a family of finite quandles.