Minimal Resolutions and the Homology of Matching and Chessboard Complexes
Victor Reiner
and Joel Roberts
DOI: 10.1023/A:1008728115910
Abstract
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: 
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 
ica and 
ivaljevi . This follows from an old observation on Betti numbers of semigroup modules over semigroup rings described 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 Björner, Lovasz, Vre ica and ivaljevi . This follows from an old observation on Betti numbers of semigroup modules over semigroup rings described in terms of simplicial complexes.Pages: 135–154
Keywords: minimal free resolution; matching complex; chessboard complex; determinantal ideal
Full Text: PDF
References
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2. K. Akin, D.A. Buchsbaum, and J. Weyman, “Schur functors and Schur complexes,” Adv. Math. 44 (1982), 207-278.
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9. Y.M. Chen, A.M. Garsia, J. Remmel, in Combinatorics and Algebra, Boulder, Colo. 1983, Algorithms for plethysm, Amer. Math. Soc., Providence, 1984, pp. 109-153.
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17. T. Józefiak, P. Pragacz, and J. Weyman, “Resolutions of determinantal varieties and tensor complexes associated with symmetric and antisymmetric matrices,” Asterisque 87-88 (1981), 109-189.
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2. K. Akin, D.A. Buchsbaum, and J. Weyman, “Schur functors and Schur complexes,” Adv. Math. 44 (1982), 207-278.
3. J. Anderson, “Determinantal rings associated with symmetric matrices: A counterexample,” Ph.D. Thesis, Univ. of Minnesota, 1992.
4. E. Babson, A. Bj\ddot orner, S. Linusson, J. Shareshian, V. Welker, “Complexes of not i-connected graphs,” MSRI preprint #1997=054, 1996.
5. A. Bj\ddot orner, L. Lovasz, S. Vrećica, and R. \check Zivaljević, “Chessboard complexes and matching complexes,” J. Lond. Math. Soc. 49 (1994), 25-39.
6. S. Bouc, “Homologie de certains ensembles de 2-sous-groups des groupes symétriques,” J. Algebra 150 (1992), 158-186.
7. W. Bruns and J. Herzog, “Semigroup rings and simplicial complexes,” Preprint 171 (Osnabr\ddot ucker Math. Schriften), Universit\ddot at Osnabr\ddot uck, 1995.
8. A. Campillo and C. Marijuan, “Higher order relations for a numerical semigroup,” Séminaire de Théorie des Nombres 3 (1991), 249-260.
9. Y.M. Chen, A.M. Garsia, J. Remmel, in Combinatorics and Algebra, Boulder, Colo. 1983, Algorithms for plethysm, Amer. Math. Soc., Providence, 1984, pp. 109-153.
10. J. Eagon and D. Northcott, “Generically acyclic complexes and generically perfect ideals,” Prod. Roy. Soc. Ser. A 299 (1967), 147-172.
11. J. Friedman and P. Hanlon, “On the Betti numbers of chessboard complexes,” preprint (1996).
12. W. Fulton and J. Harris, “Representation theory: A first course,” Graduate Texts in Mathematics: Readings in Mathematics, vol. 129,
1991. REINER AND ROBERTS
13. D. Gay, “Characters of the weyl group of SU(n) on zero weight spaces and centralizers of permutation representations,” Rocky Moun. J. Math. 6 (1976), 449-455.
14. M. Hashimoto, “Determinantal ideals without minimal free resolutions,” Nagoya J. Math. 118 (1990), 203-216.
15. M. Hashimoto, “Resolutions of determinantal ideals: t -minors of (t - 2)\times n matrices,” J. Algebra 142 (1991), 456-491.
16. M. Hochster and J.A. Eagon, “Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci,” Amer. J. Math. 93 (1971), 1020-1058.
17. T. Józefiak, P. Pragacz, and J. Weyman, “Resolutions of determinantal varieties and tensor complexes associated with symmetric and antisymmetric matrices,” Asterisque 87-88 (1981), 109-189.
18. D. Karagueusian, “Homology of complexes of degree one graphs,” Ph.D. Thesis, Stanford Univ., 1994.
19. A. Lascoux, “Syzygies des variétés determinantales,” Adv. Math. 30 (1978), 202-237.
20. P. Pragacz and J. Weyman, “Resolutions of determinantal varieties: A survey,” Lecture Notes in Mathematics Vol. 1220, pp. 73-92.
21. P. Pragacz and J. Weyman, “Complexes associated with trace and evaluation. Another approach to Lascoux' resolution,” Adv. Math. 57 (1985), 163-207.
22. R.P. Stanley, “Invariants of finite groups and their applications to combinatorics,” Bull. Amer. Math. Soc. 1 (1979), 475-511.
23. R.P. Stanley, “Linear Diophantine Equations and Local Cohomology,” Inven. Math. 68 (1982), 175-193.
24. R.P. Stanley, Combinatorics and Commutative Algebra, 2nd edition, Progress in Mathematics Vol. 41, Birkh\ddot auser, Boston, 1996.
25. B. Sturmfels, Gr\ddot obner Bases and Convex Polytopes, University Lecture Series., Vol. 8, American Mathematical Society, Providence, 1996.
26. J. Weyman, “The Grothendieck group of GL(F) \times GL(G)-equivariant modules over the coordinate ring of determinantal varieties,” preprint, 1997.
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