Homology of complexes of degree one graphs
Homology of complexes of degree one graphs
On parallel block algorithms for exact triangularizations
Parallel Computing
Exact sequences for the homology of the matching complex
Journal of Combinatorial Theory Series A
Cycle-free chessboard complexes and symmetric homology of algebras
European Journal of Combinatorics
Five-torsion in the homology of the matching complex on 14 vertices
Journal of Algebraic Combinatorics: An International Journal
Chessboard complexes indomitable
Journal of Combinatorial Theory Series A
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We generalize work of Lascoux andJózefiak-Pragacz-Weyman on Betti numbers for minimal freeresolutions of ideals generated by 2 × 2 minors of genericmatrices and generic symmetric matrices, respectively. Quotients ofpolynomial rings by these ideals are the classical Segre andquadratic Veronese subalgebras, and we compute the analogous Bettinumbers for some natural modules over these Segre and quadraticVeronese subalgebras. Our motivation is two-fold:• We immediately deduce from these results theirreducible decomposition for the symmetric group action on therational homology of all chessboard complexes and completegraph matching complexes as studied by Björner, Lovasz,Vrećica and Živaljević. This follows from an old observationon Betti numbers of semigroup modules over semigroup rings describedin terms of simplicial complexes.• The class of modules over theSegre rings and quadratic Veronese rings which we consider is closedunder the operation of taking canonical modules, and henceexposes a pleasant symmetry inherent in these Betti numbers.