Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-28T08:57:03.063Z Has data issue: false hasContentIssue false
  • English
  • Français

Character Table of a Borel Subgroup of the Ree Groups 2F4(q2)

Published online by Cambridge University Press:  01 February 2010

Frank Himstedt  and
Shih-Chang Huang
Frank Himstedt
Affiliation:
Technische Universität München, Zentrum Mathematik – M11, Boltzmannstr. 3, 85748 Garching, Germany, himstedt@ma.tum.de
Shih-Chang Huang
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand, shua003@math.auckland.ac.nz

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We compute the conjugacy classes and character table of a Borel subgroup of the Ree groups 2F4(22n+1) for all n ≥ 1 and prove that these Borel subgroups are M-groups. We determine the degrees of the irreducible characters of the Sylow-2-subgroups of 2F4(22n+1) and show that the Isaacs–Malle–Navarro version of the McKay conjecture holds for 2F4(22n+1) in characteristic 2. For most of the calculations we use CHEVIE.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 12 , 2009 , pp. 1 - 53
Copyright
Copyright © London Mathematical Society 2009

References

1.Blau, H.I. and Michler, G.O., ‘Modular representation theory of finite groups with T.I. Sylow p-subgroups’, Trans. Amer. Math. Soc. 319 (1990) 417–468.Google Scholar
2.Carter, R., Simple groups of Lie type (Wiley-Interscience, London 1972).Google Scholar
3.Carter, R., Finite groups of Lie type - Conjugacy classes and complex characters (Wiley-Interscience, London 1985).Google Scholar
4.Char, B.W., Geddes, K.O., Gonnet, G.H., Leong, B.L., Monagan, M.B., and Watt, S.M., Maple V, Language Reference Manual (Springer, 1991).CrossRefGoogle Scholar
5.Enomoto, H., and Yamada, H., ‘The characters of the finite group G 2(2n)’, Japan. J. Math 12 12 (1986) 325–377.CrossRefGoogle Scholar
6.The GAP Group, ‘GAP – Groups, Algorithms, and Programming’, Version 4.410 (2007), http://www.gap-system.org.Google Scholar
7.Geck, M., Verallgemeinerte Gelfand–Graev Charaktere und Zerlegungszahlen endlicher Gruppen von Lie-Typ, Dissertation, RWTH Aachen, 1990.Google Scholar
8.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.CrossRefGoogle Scholar
9.Himstedt, F., ‘On the 2-decomposition numbers of Steinberg's triality groups 3D 4(q), q odd’, J. Algebra 309 (2007) 569–593.CrossRefGoogle Scholar
10.Himstedt, F. and Huang, S.-C., ‘Dade's invariant conjecture for Steinberg's triality groups 3D 4(2n) in defining characteristic’, J. Algebra 316 (2007) 802–827.CrossRefGoogle Scholar
11.Huppert, B., Endliche Gruppen I, Grundlagen der mathematischen Wissenschaften (Springer, 1967).CrossRefGoogle Scholar
12.Isaacs, M., Character theory of finite groups (Dover, New York, 1976).Google Scholar
13.Isaacs, M., Malle, G. and Navarro, G., ‘A reduction theorem for the McKay conjecture’, Invent. Math. 170 (2007) 33–101.CrossRefGoogle Scholar
14.Landrock, P., Finite group algebras and their modules (Cambridge University Press, 1983).CrossRefGoogle Scholar
15.Lübeck, F., Charaktertafeln für die Gruppen CSp6(q) mit ungeradem q und Sp6(q) mit geradem q, Dissertation, Universität Heidelberg, Germany, 1993.Google Scholar
16.Malle, G., ‘Die unipotenten Charaktere von 2F 4(q 2)’, Comm. Alg. 18 (1990) 2361–2381.CrossRefGoogle Scholar
17.Ree, R., ‘A Family of simple groups associated with the simple Lie algebra of type (F 4)’, Amer. J. of Math. 83 (1961) 401–420.CrossRefGoogle Scholar
18.Shinoda, K., ‘A characterization of odd order extensions of the Ree groups 2F 4(q)’, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975) 79–102.Google Scholar
19.Shinoda, K., ‘The conjugacy classes of the finite Ree groups of type (F 4)’, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975) 1–15.Google Scholar
20.Szegedy, B., ‘Characters of the Borel and Sylow subgroups of classical groups’, J. Algebra 267 (2003) 130–136.CrossRefGoogle Scholar
21.Waki, K., ‘A note on decomposition numbers of G 2(2n)’, J. Algebra 274 (2004) 602–606.CrossRefGoogle Scholar
Supplementary material: File

JCM 12 Himstedt and Huang Appendix A

Himstedt and Huang Appendix A

Download JCM 12 Himstedt and Huang Appendix A(File)
File 62 KB