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

TL;DR

This paper characterizes the eigenvalues of irreducible representations of symmetric and alternating groups, providing a comprehensive understanding of their spectral properties in these algebraic structures.

Contribution

It offers a detailed description of the eigenvalues of permutations in irreducible representations of symmetric and alternating groups, extending previous knowledge in representation theory.

Findings

Eigenvalues of permutation representations are explicitly characterized.

Results apply to both symmetric and alternating groups.

Provides a foundation for further spectral analysis in algebraic group representations.

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.