The largest character degrees of the symmetric and alternating groups
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- by Zoltán Halasi, Carolin Hannusch and Hung Ngoc Nguyen PDF
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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 \[ b(\textsf {A}_n)^2<\sum _{\substack {\psi \in \mathrm {Irr}(\textsf {A}_n)\\ \psi (1)< b(\textsf {A}_n)}}\psi (1)^2, \] 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
- Yakov Berkovich, Groups with few characters of small degrees, Israel J. Math. 110 (1999), 325–332. MR 1750431, DOI 10.1007/BF02808187
- James P. Cossey, Zoltán Halasi, Attila Maróti, and Hung Ngoc Nguyen, On a conjecture of Gluck, Math. Z. 279 (2015), no. 3-4, 1067–1080. MR 3318260, DOI 10.1007/s00209-014-1403-6
- Christina Durfee and Sara Jensen, A bound on the order of a group having a large character degree, J. Algebra 338 (2011), 197–206. MR 2805189, DOI 10.1016/j.jalgebra.2011.04.002
- J. S. Frame, G. de B. Robinson, and R. M. Thrall, The hook graphs of the symmetric groups, Canad. J. Math. 6 (1954), 316–324. MR 62127, DOI 10.4153/cjm-1954-030-1
- The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.6.3, 2013. +http://www.gap-system.org+
- N. N. Hung, M. L. Lewis, and A. A. Schaeffer Fry, Finite groups with an irreducible character of large degree, submitted, 2015. +http://arxiv.org/abs/1505.05138+
- I. M. Isaacs, Bounding the order of a group with a large character degree, J. Algebra 348 (2011), 264–275. MR 2852240, DOI 10.1016/j.jalgebra.2011.08.037
- G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
- Michael Larsen, Gunter Malle, and Pham Huu Tiep, The largest irreducible representations of simple groups, Proc. Lond. Math. Soc. (3) 106 (2013), no. 1, 65–96. MR 3020739, DOI 10.1112/plms/pds030
- Mark L. Lewis, Bounding group orders by large character degrees: a question of Snyder, J. Group Theory 17 (2014), no. 6, 1081–1116. MR 3276228, DOI 10.1515/jgt-2014-0011
- John McKay, The largest degrees of irreducible characters of the symmetric group, Math. Comp. 30 (1976), no. 135, 624–631. MR 404414, DOI 10.1090/S0025-5718-1976-0404414-X
- Gary M. Seitz, Cross-characteristic embeddings of finite groups of Lie type, Proc. London Math. Soc. (3) 60 (1990), no. 1, 166–200. MR 1023808, DOI 10.1112/plms/s3-60.1.166
- Noah Snyder, Groups with a character of large degree, Proc. Amer. Math. Soc. 136 (2008), no. 6, 1893–1903. MR 2383494, DOI 10.1090/S0002-9939-08-09147-8
- A. M. Vershik and S. V. Kerov, Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 25–36, 96 (Russian). MR 783703
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