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

Alexey Staroletov

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

Group Theory 2025-10-03 v2 Representation Theory

Authors: Alexey Staroletov

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.

Keywords

symmetric group alternating group irreducible representation eigenvalues character theory

Cite

@article{arxiv.2501.17571,
  title  = {On eigenvalues of permutations in irreducible representations of symmetric and alternating groups},
  author = {Alexey Staroletov},
  journal= {arXiv preprint arXiv:2501.17571},
  year   = {2025}
}

Comments

An expanded version that was submitted to a journal. It also includes minor corrections. 19 pages

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