Journal of Algebraic Combinatorics

, Volume 1, Issue 1, pp 71–95

On Schur's Q-functions and the Primitive Idempotents of a Commutative Hecke Algebra

  • John R. Stembridge


Let Bn denote the centralizer of a fixed-point free involution in the symmetric group S2n. Each of the four one-dimensional representations of Bn induces a multiplicity-free representation of S2n, and thus the corresponding Hecke algebra is commutative in each case. We prove that in two of the cases, the primitive idempotents can be obtained from the power-sum expansion of Schur's Q-functions, from which follows the surprising corollary that the character tables of these two Hecke algebras are, aside from scalar multiples, the same as the nontrivial part of the character table of the spin representations of Sn.

Gelfand pairs Hecke algebras symmetric functions zonal polynomials 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. Bergeron and A.M. Garsia, “Zonal polynomials and domino tableaux,” preprint.Google Scholar
  2. 2.
    C.W. Curtis and I. Reiner, Methods of Representation Theory, Vol. I, Wiley, New York, 1981.Google Scholar
  3. 3.
    P. Diaconis, Group Representations in Probability and Statistics, Institute of Mathematical Statistics, Hayward, CA, 1988.Google Scholar
  4. 4.
    P.N. Hoffman and J.F. Humphreys, Projective representations of the symmetric groups, Oxford Univ. Press, Oxford, to appear.Google Scholar
  5. 5.
    H. Jack, “A class of symmetric functions with a parameter,” Proc. Royal Society Edinburgh Sect. A, vol. 69, pp. 1–18, 1970.Google Scholar
  6. 6.
    A.T. James, “Zonal polynomials of the real positive definite symmetric matrices,” Annals of Mathematics, vol. 74, pp. 475–501, 1961.Google Scholar
  7. 7.
    A.T. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, MA, 1981.Google Scholar
  8. 8.
    T. Józefiak, “Characters of projective representations of symmetric groups,” Expositiones Mathematicae, vol. 7, pp. 193–247, 1989.Google Scholar
  9. 9.
    T. Koornwinder, private communication.Google Scholar
  10. 10.
    D.E. Littlewood, The Theory of Group Characters, 2nd ed., Oxford University Press, Oxford, 1950.Google Scholar
  11. 11.
    I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.Google Scholar
  12. 12.
    I.G. Macdonald, “Commuting differential operators and zonal spherical functions,” in Algebraic Groups, Utrecht 1986, (A.M. Cohen et al., eds.), pp. 189–200, Lecture Notes in Mathematics, Vol. 1271, Springer-Verlag, Berlin, 1987.Google Scholar
  13. 13.
    A.O. Morris, “The spin representation of the symmetric group,” Canadian Journal of Mathematics, vol. 17, pp. 543–549, 1965.Google Scholar
  14. 14.
    J.J.C. Nimmo, “Hall-Littlewood symmetric functions and the BKP equation,” Journal of Physics A, vol. 23, pp. 751–760, 1990.Google Scholar
  15. 15.
    P. Pragacz, “Algebro-geometric applications of Schur S-and Q-polynomials,” in Séminaire d'algebre Dubreil-Malliavin 1989–90, Springer-Verlag, Berlin, to appear.Google Scholar
  16. 16.
    I. Schur, “Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen” Journal Reine Angew. Mathematics, vol. 139, pp. 155–250, 1911.Google Scholar
  17. 17.
    A.N. Sergeev, “The tensor algebra of the identity representation as a module over the Lie superalgebras gl(n,m) and Q(n),” Mathematics USSR Sbornik, vol. 51, pp. 419–427, 1985.Google Scholar
  18. 18.
    R.P Stanley, “Some combinatorial properties of Jack symmetric functions,” Advances in Mathematics, vol. 77, pp. 76–115, 1989.Google Scholar
  19. 19.
    J.R. Stembridge, “Shifted tableaux and the projective representations of symmetric groups,” Advances in Mathematics, vol. 74, pp. 87–134, 1989.Google Scholar
  20. 20.
    J.R. Stembridge, “Nonintersecting paths, pfaffians and plane partitions,” Advances in Mathematics, vol. 83, pp. 96–131, 1990.Google Scholar
  21. 21.
    J.R. Stembridge, “On symmetric functions and the spin characters of S n,” in Topics in Algebra, (S. Balcerzyk et al., eds.), Banach Center Publications, vol. 26, part 2, Polish Scientific Publishers, Warsaw, pp. 433–453, 1990.Google Scholar
  22. 22.
    Y. You, “Polynomial solutions of the BKP hierarchy and projective representations of symmetric groups,” in Infinite-Dimensional Lie Algebras and Groups, (V.G. Kac, ed.) World Scientific, Teaneck, NJ, pp. 449–464, 1989.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • John R. Stembridge
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn Arbor

Personalised recommendations