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Reducibility modulo $p$ of complex representations of finite groups of Lie type: Asymptotical result and small characteristic cases

Authors: Pham Huu Tiep and A. E. Zalesskii
Journal: Proc. Amer. Math. Soc. 130 (2002), 3177-3184
MSC (1991): Primary 20C33, 20C20
Published electronically: March 25, 2002
MathSciNet review: 1912995
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Abstract: Let $G$ be a finite group of Lie type in characteristic $p$. This paper addresses the problem of describing the irreducible complex (or $p$-adic) representations of $G$ that remain absolutely irreducible under the Brauer reduction modulo $p$. An efficient approach to solve this problem for $p > 3$has been elaborated in earlier papers by the authors. In this paper, we use arithmetical properties of character degrees to solve this problem for the groups

\begin{displaymath}G \in \{ \hspace{0.5mm}^{2}\hspace*{-0.6mm} B_{2}(q), \hspace... ...{4}(q),F_{4}(q), \hspace{0.5mm}^{3}\hspace*{-0.6mm} D_{4}(q)\} \end{displaymath}

provided that $p \leq 3$. We also prove an asymptotical result, which solves the problem for all finite groups of Lie type over ${\mathbb F}_{q}$ with $q$ large enough.

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Additional Information

Pham Huu Tiep
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105

A. E. Zalesskii
Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom

Keywords: Finite groups of Lie type, reduction modulo $p$, Steinberg representation
Received by editor(s): March 21, 2001
Received by editor(s) in revised form: June 12, 2001
Published electronically: March 25, 2002
Additional Notes: The first author was partially supported by the NSF grant DMS-0070647 and by a research award from the College of Liberal Arts and Sciences, University of Florida.
The authors are grateful to Professor J. G. Thompson and Professor B. H. Gross for constant encouragement. The authors are also thankful to the referee for helpful comments on the paper.
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2002 American Mathematical Society