On quandle representations

TL;DR

This paper characterizes when finite-dimensional quandle representations over complex numbers are decomposable or unitary, and explores their relation to the properties of the enveloping group and specific quandle families.

Contribution

It provides necessary and sufficient conditions for decomposability and unitarity of quandle representations, linking these to properties of the image elements and the enveloping group.

Findings

A quandle representation is decomposable iff all image elements are diagonalizable.

Irreducible quandle representations are unitary iff all image determinants have modulus 1.

The enveloping group admits a faithful finite-dimensional unitary representation.

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 $\rho :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 $\rho$ is diagonlizable. We show that an irreducible representation $\rho :Q \to GL(V)$ of a finite quandle over $\mathbb{C}$ is unitary for some inner product if and only if every element of the image of $\rho$ 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.