Reducibility modulo 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

DOI:
https://doi.org/10.1090/S0002-9939-02-06459-6

Published electronically:
March 25, 2002

MathSciNet review:
1912995

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Abstract: Let be a finite group of Lie type in characteristic . This paper addresses the problem of describing the irreducible complex (or -adic) representations of that remain absolutely irreducible under the Brauer reduction modulo . An efficient approach to solve this problem for 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

provided that . We also prove an asymptotical result, which solves the problem for all finite groups of Lie type over with large enough.

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Additional Information

**Pham Huu Tiep**

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

Email:
tiep@math.ufl.edu

**A. E. Zalesskii**

Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom

Email:
a.zalesskii@uea.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-02-06459-6

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