## 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.## References

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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
- MR Author ID: 733834
- 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
- MR Author ID: 843888
- Email: hungnguyen@uakron.edu
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 1947-1960 - MSC (2010): Primary 20C30, 20C15
- DOI: https://doi.org/10.1090/proc/12920
- MathSciNet review: 3460158