Minimal Resolutions and the Homology of Matching and Chessboard Complexes

We generalize work of Lascoux and Józefiak-Pragacz-Weyman on Betti numbers for minimal free resolutions of ideals generated by 2 × 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: 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, Vreica and ivaljevi. This follows from an old observation on Betti numbers of semigroup modules over semigroup rings described in terms of simplicial complexes. 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.