Lascoux-style Resolutions and the Betti Numbers of Matching and Chessboard Complexes

This paper generalizes work of Lascoux and Jo zeeak-Pragacz-Weyman computing the characteristic zero Betti numbers in minimal free resolutions of ideals generated by 2 2 minors of generic matrices and generic symmetric matrices, respectively. In the case of 2 2 minors, the quotients of certain polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we compute the analogous Betti numbers for certain natural modules over these Segre and quadratic Veronese subalgebras. The motivation for these results are twofold: Using an old observation on computing Betti numbers of semigroup modules over semigroup rings in terms of simplicial complexes, 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 Bjj orner, Lovasz, Vre cica and Zivaljevi c The class of modules over the Segre rings and quadratic Veronese rings which we consider is closed under the operation of taking canonical modules, and hence exposes a pleasant symmetry inherent in these Betti numbers.