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

### Volume 156 / 2019

Colloquium Mathematicum 156 (2019), 267-272
MSC: Primary 20G42; Secondary 22D10, 16T20.
DOI: 10.4064/cm7581-5-2018
Published online: 1 February 2019

#### Abstract

We ask whether for a given unitary representation $U$ of a compact quantum group $\mathbb G$ the associated operator $\rho_{U}\in\operatorname{Mor}(U,U^{\scriptscriptstyle\rm cc})$ has spectrum invariant under inversion; we then say that $\rho_{U}$ has symmetric eigenvalues. This is not always the case. However, we give an affirmative answer whenever a certain condition on the growth of the dimensions of irreducible subrepresentations of tensor powers of $U$ is imposed. This condition is satisfied whenever $\widehat{\mathbb G}$ is of subexponential growth.