On the Betti Numbers of Chessboard Complexes
Joel Friedman
and Phil Hanlon
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, Vre 
ica, and 
ivaljevi 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.
ica, and ivaljevi 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.Pages: 193–203
Keywords: chessboard complex; Laplacian; symmetric group; representation; connectivity; Betti number
Full Text: PDF
References
1. R. Bacher, “Valeur propre minimale du laplacien de Coxeter pour le group symétrique,” Journal of Algebra 167 (1994), 460-472.
2. A. Bj\ddot orner, L. Lovász, S.T. Vrećica, and R.T. \? Zivaljević, “Chessboard complexes and matching complexes,” J. London Math. Soc. 49 (1994), 25-39.
3. P. Diaconis, Group Representations in Probability and Statistics, Institute of Mathematical Statistics, 1988.
4. Jozef Dodziuk, “Finite-difference approach to the Hodge theory of harmonic forms,” American Journal of Mathematics, 98 (1976), 79-104.
5. J. Dodziuk and V.K. Patodi, “Riemannian structures and triangulations of manifolds,” J. of the Indian Math. Soc., 40 (1976), 1-52.
6. B. Eckmann, “Harmonische Funktionen und Randwertaufgaben in einem Komplex,” Commentarii Math. Helvetici, 17 (1944-45), 240-245.
7. L. Flatto, A.M. Odlyzko, and D.B. Wales, “Random shuffles and group representations,” Annals of Probability, 13 1985, 154-178.
8. J. Friedman, “On Cayley graphs of Sn generated by transpositions,” (to appear).
9. J. Friedman, “Computing Betti numbers via combinatorial Laplacians,” November 1995, STOC 1996 (to appear).
10. P.F. Garst, “Some Cohen-Macaulay complexes and group actions,” Ph.D. Thesis, University of Madison, Wisconsin, 1979.
11. Phil Hanlon, “A random walk on the rook placements on a Ferrer's board,” Electronic Journal of Combinatorics, (to appear).
12. W.V.D. Hodge, The Theory and Applications of Harmonic Integrals, Cambridge University Press, 1941.
13. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Welsey, 1981.
14. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Second edition, 1995.
15. James R. Munkres, Elements of Algebraic Topology, Benjamin/Cummings, 1984.
16. Edwin H. Spanier, Algebraic Topology. McGraw-Hill, 1966. (Also available from Springer-Verlag).
17. R.T. \? Zivaljević and S.T. Vrećica, “The colored Tverberg's problem and complexes of injective functions,” J. Combin. Theory Ser. A, 61 (1992), 309-318.
2. A. Bj\ddot orner, L. Lovász, S.T. Vrećica, and R.T. \? Zivaljević, “Chessboard complexes and matching complexes,” J. London Math. Soc. 49 (1994), 25-39.
3. P. Diaconis, Group Representations in Probability and Statistics, Institute of Mathematical Statistics, 1988.
4. Jozef Dodziuk, “Finite-difference approach to the Hodge theory of harmonic forms,” American Journal of Mathematics, 98 (1976), 79-104.
5. J. Dodziuk and V.K. Patodi, “Riemannian structures and triangulations of manifolds,” J. of the Indian Math. Soc., 40 (1976), 1-52.
6. B. Eckmann, “Harmonische Funktionen und Randwertaufgaben in einem Komplex,” Commentarii Math. Helvetici, 17 (1944-45), 240-245.
7. L. Flatto, A.M. Odlyzko, and D.B. Wales, “Random shuffles and group representations,” Annals of Probability, 13 1985, 154-178.
8. J. Friedman, “On Cayley graphs of Sn generated by transpositions,” (to appear).
9. J. Friedman, “Computing Betti numbers via combinatorial Laplacians,” November 1995, STOC 1996 (to appear).
10. P.F. Garst, “Some Cohen-Macaulay complexes and group actions,” Ph.D. Thesis, University of Madison, Wisconsin, 1979.
11. Phil Hanlon, “A random walk on the rook placements on a Ferrer's board,” Electronic Journal of Combinatorics, (to appear).
12. W.V.D. Hodge, The Theory and Applications of Harmonic Integrals, Cambridge University Press, 1941.
13. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Welsey, 1981.
14. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Second edition, 1995.
15. James R. Munkres, Elements of Algebraic Topology, Benjamin/Cummings, 1984.
16. Edwin H. Spanier, Algebraic Topology. McGraw-Hill, 1966. (Also available from Springer-Verlag).
17. R.T. \? Zivaljević and S.T. Vrećica, “The colored Tverberg's problem and complexes of injective functions,” J. Combin. Theory Ser. A, 61 (1992), 309-318.