Journal of Algebraic Combinatorics

, Volume 8, Issue 2, pp 193–203 | Cite as

On the Betti Numbers of Chessboard Complexes

  • Joel Friedman
  • Phil Hanlon


In this paper we study the Betti numbers of a type of simplicial complex known as a chessboard complex. We obtain a formula for their Betti numbers as a sum of terms involving partitions. This formula allows us to determine which is the first nonvanishing Betti number (aside from the 0-th Betti number). We can therefore settle certain cases of a conjecture of Björner, Lovász, Vrećica, and Živaljević in [2]. Our formula also shows that all eigenvalues of the Laplacians of the simplicial complexes are integers, and it gives a formula (involving partitions) for the multiplicities of the eigenvalues.

chessboard complex Laplacian symmetric group representation connectivity Betti number 


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  1. 1.
    R. Bacher, “Valeur propre minimale du laplacien de Coxeter pour le group symétrique,” Journal of Algebra 167 (1994), 460-472.Google Scholar
  2. 2.
    A. Björner, L. Lovász, S.T. Vrećica, and R.T. Živaljević, “Chessboard complexes and matching complexes,” J. London Math. Soc. 49 (1994), 25-39.Google Scholar
  3. 3.
    P. Diaconis, Group Representations in Probability and Statistics, Institute of Mathematical Statistics, 1988.Google Scholar
  4. 4.
    Jozef Dodziuk, “Finite-difference approach to the Hodge theory of harmonic forms,” American Journal of Mathematics, 98 (1976), 79-104.Google Scholar
  5. 5.
    J. Dodziuk and V.K. Patodi, “Riemannian structures and triangulations of manifolds,” J. of the Indian Math. Soc., 40 (1976), 1-52.Google Scholar
  6. 6.
    B. Eckmann, “Harmonische Funktionen und Randwertaufgaben in einem Komplex,” Commentarii Math. Helvetici, 17 (1944-45), 240-245.Google Scholar
  7. 7.
    L. Flatto, A.M. Odlyzko, and D.B. Wales, “Random shuffles and group representations,” Annals of Probability, 13 1985, 154-178.Google Scholar
  8. 8.
    J. Friedman, “On Cayley graphs of S n generated by transpositions,” (to appear).Google Scholar
  9. 9.
    J. Friedman, “Computing Betti numbers via combinatorial Laplacians,” November 1995, STOC 1996(to appear).Google Scholar
  10. 10.
    P.F. Garst, “Some Cohen-Macaulay complexes and group actions,” Ph.D. Thesis, University of Madison, Wisconsin, 1979.Google Scholar
  11. 11.
    Phil Hanlon, “A random walk on the rook placements on a Ferrer's board,” Electronic Journal of Combinatorics, (to appear).Google Scholar
  12. 12.
    W.V.D. Hodge, The Theory and Applications of Harmonic Integrals, Cambridge University Press, 1941.Google Scholar
  13. 13.
    G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Welsey, 1981.Google Scholar
  14. 14.
    I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Second edition, 1995.Google Scholar
  15. 15.
    James R. Munkres, Elements of Algebraic Topology, Benjamin/Cummings, 1984.Google Scholar
  16. 16.
    Edwin H. Spanier, Algebraic Topology. McGraw-Hill, 1966. (Also available from Springer-Verlag).Google Scholar
  17. 17.
    R.T. Živaljević and S.T. Vrećica, “The colored Tverberg's problem and complexes of injective functions,” J. Combin. Theory Ser. A, 61 (1992), 309-318.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Joel Friedman
    • 1
  • Phil Hanlon
    • 2
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of MathematicsUniversity of MichiganAnn Arbor

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