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Archiv der Mathematik

, Volume 103, Issue 3, pp 201–210 | Cite as

Big elements in irreducible linear groups

  • Nikolai Gordeev
  • Ulf Rehmann
Article

Abstract

Let V be a linear space over a field K of dimension n > 1, and let \({G \leq {\rm GL}(V)}\) be an irreducible linear group. In this paper we prove that the group G contains an element g such that rank \({(g - \alpha E_{n}) \geq \frac{n}{2}}\) for every \({\alpha \in K}\), where E n is the identity operator on V. This estimate is sharp for any \({n = 2^{m}}\). The existence of such an element implies that the conjugacy class of G in GL(V) intersects the big Bruhat cell \({B\dot{w}_{0}B}\) of GL(V) non-trivially (here B is a fixed Borel subgroup of G). The latter fact is equivalent to the existence of a complete flag \({\mathfrak{F}}\) such that the flags \({g(\mathfrak{F}), \mathfrak{F}}\) are in general position for some gG.

Mathematics Subject Classification

Primary 20C99 14L40 Secondary 20G05 20H20 

Keywords

Linear groups Conjugacy classes Bruhat cells Complete flags 

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References

  1. 1.
    A. Borel, Linear algebraic groups. 2nd enl.ed., Graduate texts in mathematics 126. Springer-Verlag New York Inc. 1991.Google Scholar
  2. 2.
    N. Bourbaki, Éléments de mathématique. 23. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples. Hermann, Paris 1958.Google Scholar
  3. 3.
    R. W. Carter, Finite groups of Lie type. Conjugacy Classes and Complex Characters, Reprint of the 1985 original, Wiley Classics Library, John Wiley and Sons, Chichester, 1993.Google Scholar
  4. 4.
    Yuen Chan K., Lu J.-H., Kai Ming To S.: On intersections of conjugacy classes and Bruhat cells,. Transformation groups. 15, 243–260 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    A. H. Clifford: Representations induced in an invariant subgroup,. Ann. of Math. 2(38), 533–550 (1937)Google Scholar
  6. 6.
    Ch. W. Curtis and I. Reiner, Representation theory of finite groups and associative6 algebras. Interscience Publishiers a division of John Wiley and Sons, New York, London 1962.Google Scholar
  7. 7.
    Ellers E.W., Gordeev N.: On the conjectures of J. Thompson and O. Ore,. Trans. Amer. Math. Soc. 350, 3657–3671 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ellers E.W., Gordeev N.: Intersection of conjugacy classes with Bruhat cells in Chevalley groups,. Pacific J. Math. 214, 245–261 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ellers W.E., Gordeev N.: Intersection of Conjugacy Classes with Bruhat Cells in Chevalley groups II. The cases \({SL_{n}(K), {\rm GL}_{n}(K)}\), J. Pure and Appl. Algebra, 209, 703–723 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    E. W. Ellers and N. Gordeev, Big and small elements in Chevalley groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 386 (2011), 203–226; translation in J. Math. Sci. (N. Y.) 180 (2012), 315–329.Google Scholar
  11. 11.
    Gordeev N. Sums of orbits of algebraic groups, J. Algebra 295, 62–80 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ph. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley and Sons, New York, 1978Google Scholar
  13. 13.
    Guralnick R., Kantor W. Probabilistic Generation of Finite Simple Groups, J. Algebra 234, 743–792 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Guralnick R., Malle G.: Products of conjugacy classes and fixed point spaces,. J. Amer. Math. Soc. 25, 77–121 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gordeev N., Rehmann U.: On linearly Kleiman groups. Transformation groups. 18, 685–709 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry. V1, Cambridge University Press, 1952.Google Scholar
  17. 17.
    Kawanaka N.: Unipotent elements and characters of finite Chevalley groups, Osaka J. Math. 12, 523–554 (1975)MathSciNetMATHGoogle Scholar
  18. 18.
    Lusztig G.: From conjugacy classes in the Weyl Group to unipotent classes, Represent. Theory 15, 494–530 (2011)MathSciNetMATHGoogle Scholar
  19. 19.
    Lusztig G.: On C-small conjugacy classes in a reductive group, Transform. Groups 16, 807–825 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    T. A. Springer, Linear Algebraic Groups 2nd edition, Progress in Mathematics 9. Birkhäuser Boston, Boston MA, 1998.Google Scholar
  21. 21.
    T. A. Springer and R. Steinberg, Conjugacy classes, In: A. Borel et al. Seminar on algebraic groups and related finite groups. Part E. in: Lecture Notes in Mathematics, 131. Springer-Verlag, Berlin-Heidelberg-New York, 1970.Google Scholar
  22. 22.
    Steinberg R.: Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25, 49–80 (1965)CrossRefGoogle Scholar
  23. 23.
    N. A. Vavilov, Bruhat decomposition of two-dimensional transformations, (Russian) Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 3(1989), 3–7,; translation in Vestnik Leningrad Univ. Math. 22 (1989), 1–6.Google Scholar
  24. 24.
    N. A. Vavilov and A. A. Semenov, Bruhat decomposition for long root tori in Chevalley groups, (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 175 (1989), 12–23; translation in J. Soviet Math. 57 (1991), 3453–3458.Google Scholar
  25. 25.
    N. A. Vavilov and A. A. Semenov, Long root tori in Chevalley groups, (Russian) Algebra i Analiz, 24 (2012), 22–83; translation in St.Petersburg Math. Journal. 24 (2013), 387–430.Google Scholar

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© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsRussian State Pedagogical UniversitySaint PetersburgRussia
  2. 2.Department of MathematicsBielefeld UniversityBielefeldGermany

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