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CHARACTER LEVELS AND CHARACTER BOUNDS

Part of: Probabilistic methods in group theory Representation theory of groups

Published online by Cambridge University Press:  24 January 2020

ROBERT M. GURALNICK ,
MICHAEL LARSEN  and
PHAM HUU TIEP
ROBERT M. GURALNICK
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA; guralnic@math.usc.edu
MICHAEL LARSEN
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA; mjlarsen@indiana.edu
PHAM HUU TIEP
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA; tiep@math.rutgers.edu

Abstract

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We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019, arXiv:1904.08070] by different methods.

MSC classification

Primary: 20C33: Representations of finite groups of Lie type
Secondary: 20C15: Ordinary representations and characters 20P05: Probabilistic methods in group theory
Type
Algebra
Information
Forum of Mathematics, Pi , Volume 8 , 2020 , e2
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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