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2005 | 3 | 1 | 26-38
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Quaternionic geometry of matroids

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EN
Building on a recent paper [8], here we argue that the combinatorics of matroids are intimately related to the geometry and topology of toric hyperkähler varieties. We show that just like toric varieties occupy a central role in Stanley’s proof for the necessity of McMullen’s conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperkähler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper we will give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric hyperkähler varieties. We explain how this implies the g-inequalities for rationally representable matroids. We show how the geometrical intuition in the first proof, coupled with results of Chari [3], leads to a proof of the g-inequalities for general matroid complexes, which is a recent result of Swartz [20]. The geometrical idea in the second proof will show that a pure O-sequence should satisfy the g-inequalities, thus showing that our result is in fact a consequence of a long-standing conjecture of Stanley.
Wydawca
Czasopismo
Rocznik
Tom
3
Numer
1
Strony
26-38
Opis fizyczny
Daty
wydano
2005-03-01
online
2005-03-01
Twórcy
Bibliografia
  • [1] R. Bielawski, A. Dancer: “The geometry and topology of toric hyperkähler manifolds”, Comm. Anal. Geom., Vol. 8, (2000), pp. 727–760.
  • [2] L. Billera, C. Lee: “Sufficiency of McMullen’s conditions for f-vectors of simplicial polytopes”, Bull. Amer. Math. Soc. (N.S.), Vol. 2, (1980), pp. 181–185. http://dx.doi.org/10.1090/S0273-0979-1980-14712-6
  • [3] M. Chari: “Two decompositions in topological combinatorics with applications to matroid complexes”, Trans. Amer. Math. Soc, Vol. 349, (1997), pp. 3925–3943. http://dx.doi.org/10.1090/S0002-9947-97-01921-1
  • [4] P. Griffiths, J. Harris: Principles of algebraic geometry, Wiley, New York, 1978.
  • [5] M. Harada, N. Proudfoot: “Properties of the residual circle action on a toric hyperkahler variety”, Pac. J. Math, Vol. 214, (2004), pp. 263–284. http://dx.doi.org/10.2140/pjm.2004.214.263
  • [6] W. Fulton: Introduction to Toric Varieties, Princeton University Press, New Jersey, 1993.
  • [7] T. Hausel: “Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve”, preprint, arXiv:math.AG/0406380.
  • [8] T. Hausel, B. Sturmfels: “Toric hyperkähler varieties”, Documenta Mathematica, Vol. 7, (2002), pp. 495–534, [arXiv: math.AG/0203096]
  • [9] T. Hausel, E. Swartz: “Intersection forms of toric hyperkähler varieties”, preprint, arXiv:math.AG/0306369.
  • [10] T. Hibi: “What can be said about pure O-sequences?”, J. Combin. Theory Ser. A, Vol. 50, (1989), pp. 319–322. http://dx.doi.org/10.1016/0097-3165(89)90025-3
  • [11] N. Hitchin: “The self-duality equations on a Riemann surface”, Proc. London Math. Soc., Vol. 55, (1987), pp. 59–126.
  • [12] H. Konno: “Cohomology rings of toric hyperkähler manifolds”, Internat. J. Math., Vol. 11, (2000), pp. 1001–1026. http://dx.doi.org/10.1142/S0129167X00000490
  • [13] C.M. Lopez: “Chip firing and the Tutte polynomial”,Ann. Combinatorics,Vol,1, (1997),pp. 253–259.
  • [14] P. McMullen: “The numbers of faces of simplicial polytopes”, Israel J. Math., Vol. 9, (1971), pp. 559–570.
  • [15] H. Nakajima: Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18. American Mathematical Society, Providence, RI, 1999.
  • [16] H. Nakajima: “Quiver varieties and finite-dimensional representations of quantum affine algebras”, J. Amer. Math. Soc., Vol. 14, (2001), pp. 145–238. http://dx.doi.org/10.1090/S0894-0347-00-00353-2
  • [17] R. Stanley: “The number of faces of a simplicial convex polytope”, Adv. in Math., Vol. 35, (1980), pp. 236–238. http://dx.doi.org/10.1016/0001-8708(80)90050-X
  • [18] R. Stanley: Cohen-Macaulay complexes, in Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976) Reidel, Dordrecht, 1977, pp. 51–62. NATO Adv. Study Inst. Ser., Ser. C: Math. and Phys. Sci., 31.
  • [19] R.P. Stanley: Combinatorics and Commutative Algebra, 2nd ed., Birkhäuser, Boston, 1996.
  • [20] E. Swartz: “ g-elements of matroid complexes”, Journal of Comb. Theory Ser. B, Vol. 88, (2003), pp. 369–375. http://dx.doi.org/10.1016/S0095-8956(03)00038-8
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_BF02475653
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