The largest character degrees of the symmetric and alternating groups

TL;DR

This paper establishes a bound relating the largest character degree of alternating groups to the sum of smaller degrees, confirming a prediction and answering a question in the representation theory of symmetric and alternating groups.

Contribution

It proves a new inequality bounding the largest character degree of alternating groups in terms of smaller degrees, confirming a conjecture and addressing open questions.

Findings

The largest character degree squared is less than the sum of squares of smaller degrees.

Confirms Isaacs' prediction for alternating groups.

Answers a question posed by Larsen, Malle, and Tiep.

Abstract

We show that the largest character degree of an alternating group $A_n$ with $n\geq 5$ can be bounded in terms of smaller degrees in the sense that \[ b(A_n)^2<\sum_{\psi\in\textrm{Irr}(A_n),\,\psi(1)< b(A_n)}\psi(1)^2, \] where $\textrm{Irr}(A_n)$ and $b(A_n)$ respectively denote the set of irreducible complex characters of $A_n$ and the largest degree of a character in $\textrm{Irr}(A_n)$. This confirms a prediction of I. M. Isaacs for the alternating groups and answers a question of M. Larsen, G. Malle, and P. H. Tiep.