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The largest character degrees of the symmetric and alternating groups


Authors: Zoltán Halasi, Carolin Hannusch and Hung Ngoc Nguyen
Journal: Proc. Amer. Math. Soc. 144 (2016), 1947-1960
MSC (2010): Primary 20C30, 20C15
DOI: https://doi.org/10.1090/proc/12920
Published electronically: October 14, 2015
MathSciNet review: 3460158
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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

$\displaystyle b(\textsf {A}_n)^2<\sum _{\substack {\psi \in \textup {Irr}(\textsf {A}_n)\\ \psi (1)< b(\textsf {A}_n)}}\psi (1)^2, $

where $ \textup {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 $ \textup {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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Additional Information

Zoltán Halasi
Affiliation: Department of Algebra and Number Theory, Institute of Mathematics, University of Debrecen, 4010, Debrecen, Pf. 12, Hungary
Email: halasi.zoltan@renyi.mta.hu

Carolin Hannusch
Affiliation: Department of Algebra and Number Theory, Institute of Mathematics, University of Debrecen, 4010, Debrecen, Pf. 12, Hungary
Email: carolin.hannusch@science.unideb.hu

Hung Ngoc Nguyen
Affiliation: Department of Mathematics, The University of Akron, Akron, Ohio 44325
Email: hungnguyen@uakron.edu

DOI: https://doi.org/10.1090/proc/12920
Keywords: Symmetric groups, alternating groups, character degrees, largest character
Received by editor(s): June 8, 2014
Received by editor(s) in revised form: June 23, 2015
Published electronically: October 14, 2015
Additional Notes: The research of the first author leading to these results received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 318202, from ERC Limits of discrete structures Grant No. 617747 and from OTKA K84233
The third author was partially supported by NSA Young Investigator Grant #H98230-14-1-0293 and a BCAS Faculty Scholarship Award from the Buchtel College of Arts and Sciences, The University of Akron
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society