On the Betti Numbers of Chessboard Complexes
Cite this article as: Friedman, J. & Hanlon, P. Journal of Algebraic Combinatorics (1998) 8: 193. doi:10.1023/A:1008693929682 Abstract
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 . 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 References
R. Bacher, “Valeur propre minimale du laplacien de Coxeter pour le group symétrique,”
Journal of Algebra
A. Björner, L. Lovász, S.T. Vrećica, and R.T. Živaljević, “Chessboard complexes and matching complexes,”
J. London Math. Soc.
Group Representations in Probability and Statistics, Institute of Mathematical Statistics, 1988.
Jozef Dodziuk, “Finite-difference approach to the Hodge theory of harmonic forms,”
American Journal of Mathematics
J. Dodziuk and V.K. Patodi, “Riemannian structures and triangulations of manifolds,”
J. of the Indian Math. Soc.
B. Eckmann, “Harmonische Funktionen und Randwertaufgaben in einem Komplex,”
Commentarii Math. Helvetici
L. Flatto, A.M. Odlyzko, and D.B. Wales, “Random shuffles and group representations,”
Annals of Probability
J. Friedman, “On Cayley graphs of
n generated by transpositions,” (to appear).
J. Friedman, “Computing Betti numbers via combinatorial Laplacians,” November 1995,
STOC 1996(to appear).
P.F. Garst, “Some Cohen-Macaulay complexes and group actions,” Ph.D. Thesis, University of Madison, Wisconsin, 1979.
Phil Hanlon, “A random walk on the rook placements on a Ferrer's board,”
Electronic Journal of Combinatorics, (to appear).
The Theory and Applications of Harmonic Integrals, Cambridge University Press, 1941.
G. James and A. Kerber,
The Representation Theory of the Symmetric Group, Addison-Welsey, 1981.
Symmetric Functions and Hall Polynomials, Oxford University Press, Second edition, 1995.
James R. Munkres,
Elements of Algebraic Topology, Benjamin/Cummings, 1984.
Edwin H. Spanier,
Algebraic Topology. McGraw-Hill, 1966. (Also available from Springer-Verlag).
R.T. Živaljević and S.T. Vrećica, “The colored Tverberg's problem and complexes of injective functions,”
J. Combin. Theory Ser. A
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