The largest character degrees of the symmetric and alternating groups

Halasi, Zoltán; Hannusch, Carolin; Nguyen, Hung

doi:10.1090/proc/12920

The largest character degrees of the symmetric and alternating groups
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by Zoltán Halasi, Carolin Hannusch and Hung Ngoc Nguyen PDF

Proc. Amer. Math. Soc. 144 (2016), 1947-1960 Request permission

Abstract:

We show that the largest character degree of an alternating group $\textsf {A}_n$ with $n\geq 5$ can be bounded in terms of smaller degrees in the sense that \[ b(\textsf {A}_n)^2<\sum _{\substack {\psi \in \mathrm {Irr}(\textsf {A}_n)\\ \psi (1)< b(\textsf {A}_n)}}\psi (1)^2, \] where $\mathrm {Irr}(\textsf {A}_n)$ and $b(\textsf {A}_n)$ respectively denote the set of irreducible complex characters of $\textsf {A}_n$ and the largest degree of a character in $\mathrm {Irr}(\textsf {A}_n)$. 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.

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