Symmetric Group Character Degrees and Hook Numbers

TL;DR

This paper proves that for large symmetric groups, one can find multiple disjoint sets of irreducible characters with identical degrees, each set having a unique degree, confirming a recent conjecture by Moretó.

Contribution

It establishes a combinatorial method to construct disjoint sets of characters with the same degree, resolving a conjecture about symmetric group character degrees.

Findings

Existence of k disjoint sets of j characters with same degree in large Sym(n)

Distinct sets have different degrees, confirming the conjecture

Uses duality between characters and partitions in symmetric groups

Abstract

In this article we prove the following result: that for any two natural numbers k and j, and for all sufficiently large symmetric groups Sym(n), there are k disjoint sets of j irreducible characters of Sym(n), such that each set consists of characters with the same degree, and distinct sets have different degrees. In particular, this resolves a conjecture most recently made by Moret\'o. The methods employed here are based upon the duality between irreducible characters of the symmetric groups and the partitions to which they correspond. Consequently, the paper is combinatorial in nature.