Transformation Groups

, Volume 23, Issue 1, pp 185–215 | Cite as




We prove the linkage principle and describe blocks of the general linear supergroups GL(m|n) over the ground field K of characteristic p ≠ 2.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H. H. Andersen, The strong linkage principle, J. reine angew. Math. 315 (1980), 53–59.MATHGoogle Scholar
  2. [2]
    J. Brundan, J. Kujawa, A new proof of the Mullineux conjecture, J. Algebraic Comb. 18 (2003), 13–39.CrossRefMATHGoogle Scholar
  3. [3]
    S. J. Cheng, W.Wang, Dualities and Representations of Lie Superalgebras, Graduate Studies in Mathematics, Vol. 144, American Mathematical Society, Providence, RI, 2012.Google Scholar
  4. [4]
    M. Demazure, A very simple proof of Bott's theorem, Invent. Math. 33 (1976), no. 3, 271–272.CrossRefMATHGoogle Scholar
  5. [5]
    S. Donkin, The blocks of a semisimple algebraic group, J. Algebra 67 (1980), no. 1, 36–53.CrossRefMATHGoogle Scholar
  6. [6]
    S. Doty, The strong linkage principle, Amer. J. Math. 111 (1989), no. 1, 135–141.CrossRefMATHGoogle Scholar
  7. [7]
    A. N. Grishkov, A. N. Zubkov, Solvable, reductive and quasireductive supergroups, J. Algebra 452 (2016), 448–473.CrossRefMATHGoogle Scholar
  8. [8]
    C. Gruson, V. Serganova, Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 852–892.Google Scholar
  9. [9]
    J. Germoni, Indecomposable representations of special linear Lie superalgebras, J. Algebra 209 (1998), no. 2, 367–401.CrossRefMATHGoogle Scholar
  10. [10]
    J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs, Vol. 107, American Mathematical Society, Providence, RI, 2003.Google Scholar
  11. [11]
    J. C. Jantzen, Zur Charakterformel gewisser Darstellungen halbeinfacher Gruppen und Lie-Algebren, Math. Z. 140 (1974), 127–149.CrossRefMATHGoogle Scholar
  12. [12]
    J. C. Jantzen, Darstellungen halbeinfacher Gruppen und kontravariante Formen, J. reine angew. Math. 290 (1977), 117–141.MATHGoogle Scholar
  13. [13]
    J. Kujawa, The Steinberg tensor product theorem for GL(m|n), in: Representations of Algebraic Groups, Quantum Groups, and Lie Algebras, Contemp. Math., Vol. 413, American Mathematical Society, Providence, RI, 2006, pp. 123–132.Google Scholar
  14. [14]
    J. Kujawa, Crystal structures arising from representations of GL(m|n), Represent. Theory 10 (2006), 49–85.CrossRefMATHGoogle Scholar
  15. [15]
    F. Marko, Primitive vectors in induced supermodules for general linear supergroups, J. Pure Appl. Algebra 219 (2015), no. 4, 978–1007.CrossRefMATHGoogle Scholar
  16. [16]
    F. Marko, A. N. Zubkov, Pseudocompact algebras and highest weight categories, Algebr. Represent. Theory 16 (2013), no. 3, 689–728.CrossRefMATHGoogle Scholar
  17. [17]
    И. Б. Пeнькoв, Teopия Бopeля-Beйля-Бoттa для клaccичecкиx cупepгpупп Ли. Итoги нaуки и тexн., cep. Coвpeм. пpoбл. мaт. Hoв. дocтиж., тoм 32, VINITI, M., 1988, str. 71–124. Engl. transl.: I. B. Penkov, Borel-Weil-Bott theory for classical Lie supergroups. J. Soviet Math. 51 (1990), no. 1, 2108–2140.Google Scholar
  18. [18]
    I. Penkov, I. Skornyakov, Cohomologie des D-modules tordus typiques sur les super-varits de drapeaux, C. R. Acad. Sci. Paris Sr. I Math. 299 (1984), no. 20, 1005–1008.MATHGoogle Scholar
  19. [19]
    B. Shu, W.Wang, Modular representations of the ortho-symplectic supergroups, Proc. Lond. Math. Soc. (3) 96 (2008), no. 1, 251–271.Google Scholar
  20. [20]
    D. N. Verma, Rôle of affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras, in: Lie Groups and Their Representations, I.M. Gelfand ed., Proc. Budapest 1971, London, 1975, pp. 653–705.Google Scholar
  21. [21]
    A. N. Zubkov, Some homological properties of GL(m|n) in arbitrary characteristic, J. Algebra Appl. 15 (2016), no. 7, 1650119, 26 pp.Google Scholar
  22. [22]
    A. N. Zubkov, GL(m|n)-supermodules with good and Weyl filtrations, J. Pure Appl. Algebra 219 (2015), no. 12, 5259–5279.CrossRefMATHGoogle Scholar
  23. [23]
    A. N. Zubkov, F. Marko, The center of Dist(GL(m|n)) in positive characteristic, Algebr. Represent. Theory 19 (2016), no. 3, 613–639.CrossRefMATHGoogle Scholar
  24. [24]
    A. N. Zubkov, Affine quotients of supergroups, Transform. Groups 14 (2009), no. 3, 713–745.CrossRefMATHGoogle Scholar
  25. [25]
    A. H. Зубкoв, O нeкoтopыx cвoйcтвax oбщиx линeйныx cупepгpупп и cупepaлгeбp Шуpa, Aлгeбpa и лoгикa 45 (2006), no. 3, 257–299. Engl. transl.: A. N. Zubkov, Some properties of general linear supergroups and of Schur superalgebras, Algebra Logic 45 (2006), no. 3, 147–171.Google Scholar
  26. [26]
    A. N. Zubkov, On quotients of affine superschemes over finite supergroups, J. Algebra Appl. 10 (2011), no. 3, 391–408.CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.The Pennsylvania State UniversityHazletonUSA
  2. 2.Sobolev Institute of Mathematics Siberian Branch of The Russian Academy of SciencesNovosibirskRussia
  3. 3.Omsk State Technical UniversityOmskRussia

Personalised recommendations