The minimal polynomials of powers of cycles in the ordinary representations of symmetric and alternating groups

TL;DR

This paper determines the minimal polynomials of certain permutations in the irreducible representations of symmetric and alternating groups, focusing on permutations with cycles of equal length.

Contribution

It provides explicit formulas for minimal polynomials of powers of cycles in the ordinary irreducible representations of $A_n$ and $S_n$, a novel result in representation theory.

Findings

Explicit minimal polynomials for permutations with equal-length cycles.

Results applicable to both symmetric and alternating groups.

Enhances understanding of permutation representations in algebra.

Abstract

Denote the alternating and symmetric groups of degree $n$ by $A_n$ and $S_n$ respectively. Consider a permutation $\sigma\in S_n$ all of whose nontrivial cycles are of the same length. We find the minimal polynomials of $\sigma$ in the ordinary irreducible representations of $A_n$ and $S_n$.