On Minimal Polynomials of Elements in Symmetric and Alternating Groups

TL;DR

This paper characterizes the minimal polynomials of elements in symmetric and alternating groups' irreducible representations, providing new proofs of classical theorems and exploring eigenvalues of these elements.

Contribution

It completely describes minimal polynomials of group elements in irreducible representations, offering new combinatorial proofs of existing theorems and analyzing eigenvalues for elements of even order.

Findings

Minimal polynomial often equals x^{o(g)} - 1 with explicit exceptions.

New combinatorial proof of Swanson's theorem on Young tableaux.

Most elements of even order have eigenvalue -1, with some exceptions.

Abstract

Let $ (\rho, V) $ be an irreducible representation of the symmetric group $ S_n$ (or the alternating group $ A_n$), and let $ g $ be a permutation on $n$ letters with each of its cycle lengths divides the length of its largest cycle. We describe completely the minimal polynomial of $\rho(g)$, showing that, in most cases, it equals $x^{o(g)} - 1 $, with a few explicit exceptions. As a by-product, we obtain a new proof (using only combinatorics and representation theory) of a theorem of Swanson that gives a necessary and sufficient condition for the existence of a standard Young tableau of a given shape and major index $r \ \text{mod} \ n$, for all $r$. Thereby, we give a new proof of a celebrated result of Klyachko on Lie elements in a tensor algebra, and of a conjecture of Sundaram on the existence of an invariant vector for $n$-cycles. We also show that for elements $g$ in $S_n$ or $A_n$ of even order, in most cases, $\rho(g)$ has eigenvalue $-1$, with a few explicit exceptions.