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September 2018 Character bounds for finite groups of Lie type
Roman Bezrukavnikov, Martin W. Liebeck, Aner Shalev, Pham Huu Tiep
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Acta Math. 221(1): 1-57 (September 2018). DOI: 10.4310/ACTA.2018.v221.n1.a1


We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form $\lvert \chi(g) \rvert \leqslant c \chi (1)^{\alpha g}$, and give rise to a variety of applications, for example to covering numbers and mixing times of random walks on such groups. In particular, we deduce that, if $G$ is a classical group in dimension $n$, then, under some conditions on $G$ and $g \in G$, the mixing time of the random walk on $G$ with the conjugacy class of $g$ as a generating set is (up to a small multiplicative constant) $n/s$, where $s$ is the support of $g$.

Funding Statement

The first author was partially supported by the NSF grants DMS-1102434 and DMS-1601953. The second and third authors acknowledge the support of EPSRC grant EP/H018891/1. The third author acknowledges the support of ERC advanced grant 247034, ISF grants 1117/13 and 686/17, BSF grant 2016072 and the Vinik chair of mathematics which he holds. The fourth author was partially supported by the NSF grants DMS-1839351 and DMS-1840702, the Simons Foundation Fellowship 305247, the EPSRC, and the Mathematisches Forschungsinstitut Oberwolfach.


Download Citation

Roman Bezrukavnikov. Martin W. Liebeck. Aner Shalev. Pham Huu Tiep. "Character bounds for finite groups of Lie type." Acta Math. 221 (1) 1 - 57, September 2018.


Received: 2 November 2017; Revised: 7 August 2018; Published: September 2018
First available in Project Euclid: 19 June 2019

zbMATH: 06983623
MathSciNet: MR3877017
Digital Object Identifier: 10.4310/ACTA.2018.v221.n1.a1

Rights: Copyright © 2018 Institut Mittag-Leffler


Vol.221 • No. 1 • September 2018
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