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

Abstract

Denote the alternating and symmetric groups of degree [Formula: see text] by [Formula: see text] and [Formula: see text], respectively. Consider a permutation [Formula: see text], all of whose nontrivial cycles are of the same length. We find the minimal polynomials of [Formula: see text] in the ordinary irreducible representations of [Formula: see text] and [Formula: see text].