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

TL;DR

The paper provides a complete description of the eigenvalues of irreducible $S_n$-representations on permutations by associating each pair $(λ,μ)$ to a minimal polynomial $p_λ^μ(x)$ and spectrum $Sp_λ(μ)$, with explicit rules for when $p_λ^μ(x)\neq x^{o(σ)}-1$. The method relies on the Littlewood–Richardson rule, branching laws, and a careful induction on the degree, augmented by corner-removal techniques and LR-tableaux to control spectral factors. The results extend to $A_n$, where irreducibles either remain irreducible or split into pairs $χ_λ^{±}$, yielding analogous eigenvalue descriptions for permutations with cycle-type $μ$. Together, these findings provide cycle-structure–driven, combinatorial formulas for eigenvalues and minimal polynomials in both symmetric and alternating groups, linking representation theory with tableau combinatorics. The work deepens understanding of spectral behavior of permutation elements across classical families of finite groups and offers explicit tools for future algebraic and computational investigations.

Abstract

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

Figures (5)

• Figure 1: Cases of diagrams before the last step
• Figure 2: Tableaux for $\lambda=(n-2,1,1)$
• Figure 3: Examples of $\operatorname{LR}$-tableaux
• Figure 4: Case $\lambda_1-\lambda_2\geqslant d_1$
• Figure 5: Cases $\gamma\in\{(2,2),(2,2,2),(3,3)\}$

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof