Algebras and Representation Theory

, Volume 22, Issue 5, pp 1133–1147 | Cite as

Bases for Spaces of Highest Weight Vectors in Arbitrary Characteristic

  • Adam Dent
  • Rudolf TangeEmail author
Open Access


Let k be an algebraically closed field of arbitrary characteristic. First we give explicit bases for the highest weight vectors for the action of GLr ×GLs on the coordinate ring \(k[\text {Mat}_{rs}^{m}]\) of m-tuples of r × s-matrices. It turns out that this is done most conveniently by giving an explicit good GLr ×GLs-filtration on \(k[\text {Mat}_{rs}^{m}]\). Then we deduce from this result explicit spanning sets of the \(k[\text {Mat}_{n}]^{\text {GL}_{n}}\)-modules of highest weight vectors in the coordinate ring k[Matn] under the conjugation action of GLn.


Highest weight vector 

Mathematics Subject Classification (2010)

13A50 16W22 20G05 


  1. 1.
    Akin, K., Buchsbaum, D.A., Weyman, J.: Schur functors and Schur complexes. Adv. Math. 44(3), 207–278 (1982)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Clausen, M.: Letter place algebras and a characteristic-free approach to the representation theory of the general linear and symmetric groups. I., Adv. Math. 33 (2), 161–191 (1979)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Clausen, M.: Letter place algebras and a characteristic-free approach to the representation theory of the general linear and symmetric groups. II., Adv. Math. 38 (2), 152–177 (1980)MathSciNetCrossRefGoogle Scholar
  4. 4.
    De Concini, C., Eisenbud, D., Procesi, C.: Young diagrams and determinantal varieties. Invent. Math. 56(2), 129–165 (1980)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Désarménien, J., Kung, J.P.S., Rota, G.-C.: Invariant theory, Young bitableaux, and combinatorics. Adv. Math. 27(1), 63–92 (1978)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Donin, I.F.: Skew diagrams and decomposition into irreducible components of exterior and symmetric powers of the adjoint representation of GL(n) and GL(p/q), Reports Department of Mathematics University of Stockholm, Seminar on supermanifolds, Vol. 8Google Scholar
  7. 7.
    Donin, I.F.: Decompositions of tensor products of representations of a symmetric group and of symmetric and exterior powers of the adjoint representation of gl(N). Soviet Math. Dokl. 38(3), 654–658 (1989)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Donkin, S.: Skew modules for reductive groups. J. Algebra 113(2), 465–479 (1988)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Donkin, S.: On tilting modules for algebraic groups. Math. Z. 212(1), 39–60 (1993)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Green, J.A.: Polynomial Representations of GLN, Corrected and Augmented Edition. Lecture Notes in Mathematics, vol. 830. Springer, Berlin (2007)Google Scholar
  11. 11.
    Kouwenhoven, F.: schur and Weyl functors. Adv. Math. 90(1), 77–113 (1991)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Macdonald, I.G.: Symmetric functions and Hall polynomials, 2nd edn. Oxford University Press, New York (1995)Google Scholar
  13. 13.
    James, G.D., Peel, M.H.: Specht series for skew representations of symmetric groups. J. Algebra 56(2), 343–364 (1979)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jantzen, J.C.: Representations of algebraic groups, 2nd edn. American Mathematical Society, Providence (2003)CrossRefGoogle Scholar
  15. 15.
    Tange, R.: Transmutation and highest weight vectors. to appear in Transform. GroupsGoogle Scholar
  16. 16.
    Zelevinsky, A.V.: A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence. J. Algebra 69(1), 82–94 (1981)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zelevinsky, A.V.: Representations of Finite Classical Groups. A Hopf Algebra approach. Lecture Notes in Mathematics, vol. 869. Springer-Verlag, Berlin-New York (1981)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of MathematicsUniversity of LeedsLeedsUK

Personalised recommendations