Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T21:23:42.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Low-dimensional representations of finite orthogonal groups

Part of: Linear algebraic groups and related topics Representation theory of groups

Published online by Cambridge University Press:  26 January 2021

KAY MAGAARD  and
GUNTER MALLE
KAY MAGAARD
Affiliation:
FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653Kaiserslautern, Germany.
GUNTER MALLE*
Affiliation:
FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653Kaiserslautern, Germany.
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We determine the smallest irreducible Brauer characters for finite quasi-simple orthogonal type groups in non-defining characteristic. Under some restrictions on the characteristic we also prove a gap result showing that the next larger irreducible Brauer characters have a degree roughly the square of those of the smallest non-trivial characters.

MSC classification

Primary: 20C33: Representations of finite groups of Lie type
Secondary: 20D06: Simple groups 20G40: Linear algebraic groups over finite fields
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 3 , November 2021 , pp. 585 - 606
Copyright
© Cambridge Philosophical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bonnafé, C., Dat, J.-F. and Rouquier, R.. Derived categories and Deligne–Lusztig varieties II. Ann. of Math. (2) 185 (2017), 609–670.Google Scholar
Cabanes, M. and Enguehard, M.. Representation Theory of Finite Reductive Groups. (Cambridge University Press, Cambridge, 2004).Google Scholar
Dudas, O. and Malle, G.. Bounding Harish–Chandra series. Trans. Amer. Math. Soc. 371 (2019), 6511–6530.Google Scholar
Geck, M., Hiss, G., Lübeck, F., Malle, G. and Pfeiffer, G.. CHEVIE — A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras. Appl. Algebra Engrg. Comm. Comput. 7 (1996), 175–210.Google Scholar
Geck, M. and Malle, G.. The Character Theory of Finite Groups of Lie Type: A Guided Tour. (Cambridge University Press, Cambridge, 2020).Google Scholar
Gruber, J. and Hiss, G.. Decomposition numbers of finite classical groups for linear primes. J. Reine Angew. Math. 485 (1997), 55–91.Google Scholar
Guralnick, R. M., Magaard, K., Saxl, J. and Tiep, P. H.. Cross characteristic representations of symplectic and unitary groups. J. Algebra 257 (2002), 291–347.10.1016/S0021-8693(02)00527-6CrossRefGoogle Scholar
Guralnick, R. M. and Tiep, P. H.. Cross characteristic representations of even characteristic symplectic groups. Trans. Amer. Math. Soc. 356 (2004), 4969–5023.10.1090/S0002-9947-04-03477-4CrossRefGoogle Scholar
Hiss, G. and Malle, G.. Low-dimensional representations of special unitary groups. J. Algebra 236 (2001), 745–767.Google Scholar
Lübeck, F.. Character degrees and their multiplicities for some groups of Lie type of rank < 9. Available at http://www.math.rwth-aachen.de/˜Frank.Luebeck/chev/DegMult/.Google Scholar
Magaard, K., Röhrle, G. and Testerman, D. M.. On the irreducibility of symmetrizations of cross-characteristic representations of finite classical groups. J. Pure Appl. Algebra 217 (2013), 1427–1446.10.1016/j.jpaa.2012.11.004CrossRefGoogle Scholar
Nguyen, H. N.. Low-dimensional complex characters of the symplectic and orthogonal groups. Comm. Algebra 38 (2010), 1157–1197.10.1080/00927870902953989CrossRefGoogle Scholar
Tiep, P. H. and Zalesskii, A.. Minimal characters of the finite classical groups. Comm. Algebra 24 (1996), 2093–2167.10.1080/00927879608825690CrossRefGoogle Scholar
White, D.. Decomposition numbers of Sp (4, q) for primes dividing q ± 1. J. Algebra 132 (1990), 488–500.10.1016/0021-8693(90)90143-CCrossRefGoogle Scholar