On the irreducible character degrees of symmetric groups and their multiplicities

TL;DR

This paper investigates the largest irreducible character degrees of symmetric groups, presents new conjectures based on computational evidence, and proves results about the multiplicities and restrictions of these degrees for large n.

Contribution

It introduces new conjectures on character degree multiplicities, proves a specific lower bound for symmetric groups with n≥21, and explores related questions for unipotent degrees.

Findings

At least eight irreducible characters of S_n have the same degree for n≥21.

Computational experiments support new conjectures on character degree multiplicities.

Insights into algorithms for finding the largest irreducible character degree of S_n.

Abstract

We consider problems concerning the largest degrees of irreducible characters of symmetric groups, and the multiplicities of character degrees of symmetric groups. Using evidence from computer experiments, we posit several new conjectures or extensions of previous conjectures, and prove a number of results. One of these is that, if $n\geq 21$, then there are at least eight irreducible characters of $S_n$, all of which have the same degree, and which have irreducible restriction to $A_n$. We explore similar questions about unipotent degrees of $\mathrm{GL}_n(q)$. We also make some remarks about how the experiments here shed light on posited algorithms for finding the largest irreducible character degree of $S_n$.