Symmetry of eigenvalues of operators associated with representations of compact quantum groups

TL;DR

This paper investigates when operators linked to unitary representations of compact quantum groups have symmetric eigenvalues, providing conditions related to the growth of irreducible subrepresentations that ensure this symmetry.

Contribution

The paper establishes that operators associated with certain unitary representations have symmetric eigenvalues under specific growth conditions of subrepresentations.

Findings

Eigenvalues are symmetric under certain conditions.

Subexponential growth of dual quantum group implies eigenvalue symmetry.

Provides criteria for spectral symmetry in quantum group representations.

Abstract

We ask the question whether for a given unitary representation $U$ the associated operator $\rho_{U}\in\operatorname{Mor}(U,U^{c\, c})$ has spectrum invariant under inversion -- in this case we say that $\rho_{U}$ has symmetric eigenvalues. This does not have to be the case. However, we give affirmative answer whenever a certain condition on the growth of dimensions of irreducible subrepresentations of tensor powers of $U$ is imposed. This condition is satisfied whenever $\widehat{\mathbb{G}}$ is of subexponential growth.