Geodesic
On the Betti numbers of chessboard complexes
Journal of Algebraic Combinatorics, Tome 8 (1998) no. 2, pp. 193-203.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $cacute$ica, and $Zcaron$ivaljevi $cacute$ 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.
Keywords: chessboard complex, Laplacian, symmetric group, representation, connectivity, Betti number 
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     title = {On the {Betti} numbers of chessboard complexes},
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Friedman, Joel; Hanlon, Phil. On the Betti numbers of chessboard complexes. Journal of Algebraic Combinatorics, Tome 8 (1998) no. 2, pp. 193-203. http://geodesic.mathdoc.fr/item/JAC_1998__8_2_a1/