Advertisement

On the Number of Characters in Blocks of Quasi-simple Groups

  • Gunter MalleEmail author
Article
  • 5 Downloads

Abstract

We prove, for primes p ≥ 5, two inequalities between the fundamental invariants of Brauer p-blocks of finite quasi-simple groups: the number of characters in the block, the number of modular characters, the number of height 0 characters, and the number of conjugacy classes of a defect group and of its derived subgroup. For this, we determine these invariants explicitly, or at least give bounds for them for several classes of classical groups.

Keywords

Number of simple modules Invariants of blocks Inequalities for blocks of simple groups 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

I thank Frank Himstedt for providing me with information on the principal 2- and 3-blocks of 3D4(q), Donna Testerman for a careful reading of a preliminary version, as well as Zhicheng Feng and Sofia Brenner for some remarks and the anonymous referee for his help with the proper use of the English language and for pointing out three inaccuracies in a previous version of Table 2.

References

  1. 1.
    Bonnafé, C., Dat, J.-F., Rouquier, R.: Derived categories and Deligne–Lusztig varieties II. Ann. Math. 185(2), 609–670 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Broué, M., Malle, G., Michel, J.: Generic blocks of finite reductive groups. Astérisque 212, 7–92 (1993)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Brunat, O., Lübeck, F.: On defining characteristic representations of finite reductive groups. J. Algebra 395, 121–141 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cabanes, M.: Odd character degrees for Sp(2n, 2). C. R. Math. Acad. Sci. Paris 349, 611–614 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cabanes, M., Enguehard, M.: On unipotent blocks and their ordinary characters. Invent. Math. 117, 149–164 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cabanes, M., Enguehard, M.: Representation Theory of Finite Reductive Groups. New Mathematical Monographs, vol. 1. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  7. 7.
    Denoncin, D.: Stable basic sets for finite special linear and unitary groups. Adv. Math. 307, 344–368 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Deriziotis, D., Michler, G.: Character table and blocks of finite simple triality groups 3 D 4(q). Trans. Amer. Math. Soc. 303, 39–70 (1987)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Enguehard, M.: Sur les l-blocs unipotents des groupes réductifs finis quand l est mauvais. J. Algebra 230, 334–377 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Enguehard, M.: Vers une décomposition de Jordan des blocs des groupes réductifs finis. J. Algebra 319, 1035–1115 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fulman, J., Guralnick, R.: Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements. Trans. Amer. Math. Soc. 364, 3023–3070 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Geck, M., Hiss, G.: Basic sets of Brauer characters of finite groups of Lie type. J. Reine Angew. Math. 418, 173–188 (1991)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Himstedt, F.: On the 2-decomposition numbers of Steinberg’s triality groups 3 D 4(q), q odd. J. Algebra 309, 569–593 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Himstedt, F., Huang, S.: On the decomposition numbers of Steinberg’s triality groups 3 D 4(2n) in odd characteristics. Comm. Algebra 41, 1484–1498 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hiss, G., Shamash, J.: 3-blocks and 3-modular characters of G 2(q). J. Algebra 131, 371–387 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hiss, G., Shamash, J.: 2-blocks and 2-modular characters of the Chevalley groups G 2(q). Math. Comp. 59, 645–672 (1992)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Howlett, R.B.: On the degrees of Steinberg characters of Chevalley groups. Math. Z. 135(74), 125–135 (1973)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kessar, R., Malle, G.: Quasi-isolated blocks and Brauer’s height zero conjecture. Ann. Math. 178(2), 321–384 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kleshchev, A., Tiep, P.H.: Representations of finite special linear groups in non-defining characteristic. Adv. Math. 220, 478–504 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Malle, G.: Die unipotenten Charaktere von 2 F 4(q 2). Comm. Algebra 18, 2361–2381 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Malle, G.: On a minimal counterexample to Brauer’s k(B)-conjecture. Israel J. Math. 228, 527–556 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Malle, G., Navarro, G.: Inequalities for some blocks of finite groups. Archiv Math. 87, 390–399 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Malle, G., Testerman, D.: Linear Algebraic Groups and Finite Groups of Lie Type. Cambridge Studies in Advanced Mathematics, vol. 133. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  24. 24.
    Michler, G.O., Olsson, J.B.: Character correspondences in finite general linear, unitary and symmetric groups. Math. Z. 184, 203–233 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Navarro, G.: Some remarks on global/local conjectures. In: Finite Simple Groups: Thirty Years of the Atlas and Beyond, Contemp Math, vol. 694, pp 151–158. Amer. Math. Soc., Providence (2017)Google Scholar
  26. 26.
    Olsson, J.B.: McKay numbers and heights of characters. Math. Scand. 38, 25–42 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Olsson, J.: On the number of characters in blocks of finite general linear, unitary and symmetric groups. Math. Z. 186, 41–47 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Olsson, J.B.: On the p-blocks of symmetric and alternating groups and their covering groups. J. Algebra 128, 188–213 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Olsson, J.B.: Combinatorics and Representations of Finite Groups. Universität Essen, Fachbereich Mathematik, Essen (1993)zbMATHGoogle Scholar
  30. 30.
    Pantea, C.A.: On the number of conjugacy classes of finite p-groups. Mathematica 46(69), 193–203 (2004)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Sambale, B.: Blocks of Finite Groups and their Invariants. Lecture Notes in Mathematics, vol. 2127. Springer, Cham (2014)Google Scholar
  32. 32.
    The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.4. http://www.gap-system.org (2004)

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.FB MathematikTU KaiserslauternKaiserslauternGermany

Personalised recommendations