On eigenvalues of permutations in irreducible representations of symmetric and alternating groups

Abstract

Denote the symmetric group of degree $n$ by $S_n$. Let $\rho$ be an irreducible representation of $S_n$ over the field of complex numbers and $\sigma\in S_n$. In this paper, we describe the set of eigenvalues of $\rho(\sigma)$. Based on this result, we also obtain a description in the case of alternating groups.