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

Abstract

We ask the question whether for a given unitary representation $U$ the associated operator ${\rho }_U\in \mathrm{Mor}(U,U^{cc})$ 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 $\hat{\mathbb{G}}$ is of subexponential growth.

5 pages