Advertisement

Sums of squares of Littlewood–Richardson coefficients and GLn-harmonic polynomials

  • Pamela E. HarrisEmail author
  • Jeb F. Willenbring
Chapter
Part of the Progress in Mathematics book series (PM, volume 257)

Abstract

We consider the example from invariant theory concerning the conjugation action of the general linear group on several copies of the n × n matrices, and examine a symmetric function which stably describes the Hilbert series for the invariant ring with respect to the multigradation by degree. The terms of this Hilbert series may be described as a sum of squares of Littlewood–Richardson coefficients. A “principal specialization” of the gradation is then related to the Hilbert series of the K-invariant subspace in the GL n -harmonic polynomials (in the sense of Kostant), where K denotes a block diagonal embedding of a product of general linear groups. We also consider other specializations of this Hilbert series.

Keywords

Littlewood–Richardson coefficients Harmonic functions 

Mathematics Subject Classification

05E05 17B10 22E46 

References

  1. 1.
    David Benson, Walter Feit, and Roger Howe, Finite linear groups, the Commodore 64, Euler and Sylvester. Amer. Math. Monthly, 93(9):717–719, 1986.Google Scholar
  2. 2.
    C. Chevalley, Sur certains groupes simples. Tôhoku Math. J. (2), 7:14–66, 1955.Google Scholar
  3. 3.
    Vesselin Drensky, Computing with matrix invariants. Math. Balkanica (N.S.), 21(1–2):141–172, 2007.Google Scholar
  4. 4.
    William Fulton, Young tableaux, Vol. 35, London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry.Google Scholar
  5. 5.
    Roe Goodman and Nolan R. Wallach, Symmetry, Representations, and Invariants, Vol. 255, Graduate Texts in Mathematics. Springer, New York, 2009.Google Scholar
  6. 6.
    Timothy Gowers, June Barrow-Green, and Imre Leader (editors), The Princeton Companion to Mathematics. Princeton University Press, Princeton, NJ, 2008.Google Scholar
  7. 7.
    Pamela E. Harris, On the adjoint representation of \(\mathfrak{s}\mathfrak{l}_{n}\) and the Fibonacci numbers. C. R. Math. Acad. Sci. Paris, 349(17–18):935–937, 2011.MathSciNetzbMATHGoogle Scholar
  8. 8.
    W. H. Hesselink, Characters of the nullcone. Math. Ann., 252(3):179–182, 1980.CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Roger Howe and Soo Teck Lee, Bases for some reciprocity algebras. I. Trans. Amer. Math. Soc., 359(9):4359–4387, 2007.Google Scholar
  10. 10.
    Roger Howe and Soo Teck Lee, Why should the Littlewood-Richardson rule be true? Bull. Amer. Math. Soc. (N.S.), 49(2):187–236, 2012.Google Scholar
  11. 11.
    Roger Howe, Eng-Chye Tan, and Jeb F. Willenbring, Stable branching rules for classical symmetric pairs. Trans. Amer. Math. Soc., 357(4):1601–1626, 2005.Google Scholar
  12. 12.
    Bertram Kostant, Lie group representations on polynomial rings. Amer. J. Math., 85:327–404, 1963.Google Scholar
  13. 13.
    I. G. Macdonald, Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications.Google Scholar
  14. 14.
    C. Procesi, The invariant theory of n × n matrices. Advances in Math., 19(3):306–381, 1976.CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Claudio Procesi, The invariants of n × n matrices. Bull. Amer. Math. Soc., 82(6):891–892, 1976.Google Scholar
  16. 16.
    Richard Stong, Some asymptotic results on finite vector spaces. Adv. in Appl. Math., 9(2):167–199, 1988.Google Scholar
  17. 17.
    N. R. Wallach and J. Willenbring, On some q-analogs of a theorem of Kostant-Rallis. Canad. J. Math., 52(2):438–448, 2000.CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Jeb F. Willenbring, Stable Hilbert series of \(\mathcal{S}(\mathfrak{g})^{K}\) for classical groups. J. Algebra, 314(2): 844–871, 2007.CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.MSCS DepartmentMarquette UniversityMilwaukeeUSA
  2. 2.Department of Mathematical SciencesUniversity of Wisconsin-MilwaukeeMilwaukeeUSA

Personalised recommendations