Archiv der Mathematik

, Volume 110, Issue 5, pp 433–446 | Cite as

Singer cycles in 2-modular representations of the group \(GL_{n+1}(2)\)

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Abstract

We determine the irreducible 2-modular representations of the group \(G=GL_{n+1}(2)\) in which a Singer cycle has eigenvalue 1, and show that in these representations every element \(g\in G\) has eigenvalue 1.

Keywords

Singer cycles General linear group Modular representations Eigenvalue 1 

Mathematics Subject Classification

20G05 20G40 

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References

  1. 1.
    N. Bourbaki, Éléments de mathématique. Fast. XXXIV: Groupes et algebres de Lie, Ch. IV-VI, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968, 288 pp.Google Scholar
  2. 2.
    N. Bourbaki, Éléments de mathématique. Fast. XXXVIII: Groupes et algebres de Lie, Ch. VII-VIII, Actualités Scientifiques et Industrielles, No. 1364, Hermann, Paris, 1975, 271 pp.Google Scholar
  3. 3.
    R. Guralnick and P. H. Tiep, Finite simple unisingular groups of Lie type, J. Group Theory 6 (2003), 271–310.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    J. Humphreys, Modular Representations of Finite Groups of Lie Type, London Mathematical Society, Lecture Note Series, 326, Cambridge Univ. Press, Cambridge, 2006.Google Scholar
  5. 5.
    A.S. Kondrat‘ev, O. Osinovskaia, and I.D. Suprunenko, On the behavior of elements of prime order from Zinger cycles in representations of a special linear group, Proc. Steklov Inst. Math. 285 (2014), 108–115.Google Scholar
  6. 6.
    P. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Mathematical Society, Lecture Note Series, 126, Cambridge Univ. Press, Cambridge, 1990.CrossRefMATHGoogle Scholar
  7. 7.
    I.D. Suprunenko, Preservation of the weight systems of irreducible representations of algebraic groups and Lie algebras of type \(A\) with restricted highest weights under reduction modulo \(p\) (in Russian), Vesti Akad. Navuk BSSR, Ser. Fiz.-Mat. Navuk (1983), 18–22.Google Scholar
  8. 8.
    I.D. Suprunenko and A.E. Zalesski, Fixed vectors for elements in representations of algebraic groups, Internat. J. Algebra Comput. 17 (2007), 1249–1261.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    A.E. Zalesski and P. H. Tiep, Hall-Higman type theorems for semisimple elements of finite classical groups, Proc. London Math. Soc. 97 (208), 623–668.Google Scholar
  10. 10.
    A.E. Zalesski, The eigenvalue \(1\) of matrices of complex representations of finite Chevalley groups, Proc. Steklov Inst. Math. (1991), 109–119.Google Scholar
  11. 11.
    A.E. Zalesski, Singer tori in irreducible representations of \(GL(n,q)\), J. Group Theory 19 (2016), 523–542.MathSciNetMATHGoogle Scholar
  12. 12.
    A.E. Zalesski, Invariants of maximal tori and unipotent constituents of some quasi-projective characters for finite classical groups, J. Algebra 500(2018), 517–541.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Physics, Mathematics and InformaticsNational Academy of Sciences of BelarusMinskBelarus

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