The largest character degrees of the symmetric and alternating groups

Abstract

We show that the largest character degree of an alternating group $A_n$ with $n\ge 5$ can be bounded in terms of smaller degrees in the sense that \[ b(A_n)^2<\sum_{ψ\in\textrm{Irr}(A_n),\,ψ(1)< b(A_n)}ψ(1)^2, \] where $\text{Irr}(A_n)$ and $b(A_n)$ respectively denote the set of irreducible complex characters of $A_n$ and the largest degree of a character in $\text{Irr}(A_n)$. 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.

Cited by in corpus (3)

• Irreducible characters of even degree and normal Sylow $2$-subgroups
• Finite groups with an irreducible character of large degree
• On the second eigenvalue of a Cayley graph of the symmetric group