Geodesic
Minimal resolutions and the homology of matching and chessboard complexes
Journal of Algebraic Combinatorics, Tome 11 (2000) no. 2, pp. 135-154.

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

Summary: We generalize work of Lascoux and Józefiak-Pragacz-Weyman on Betti numbers for minimal free resolutions of ideals generated by $2 \times 2$ minors of generic matrices and generic symmetric matrices, respectively. Quotients of polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we compute the analogous Betti numbers for some natural modules over these Segre and quadratic Veronese subalgebras. Our motivation is two-fold: $bull$ We immediately deduce from these results the irreducible decomposition for the symmetric group action on the rational homology of all chessboard complexes and complete graph matching complexes as studied by Björner, Lovasz, Vre $cacute$ica and $Zcaron$ivaljevi $cacute$. This follows from an old observation on Betti numbers of semigroup modules over semigroup rings described in terms of simplicial complexes.
Keywords: minimal free resolution, matching complex, chessboard complex, determinantal ideal 
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Reiner, Victor; Roberts, Joel. Minimal resolutions and the homology of matching and chessboard complexes. Journal of Algebraic Combinatorics, Tome 11 (2000) no. 2, pp. 135-154. http://geodesic.mathdoc.fr/item/JAC_2000__11_2_a1/