Abstract
We study Stanley’s long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable artinian level monomial ideals, as often done in the past, to the study of properties of pure O-sequences. We propose a conjecture on pure O-sequences and settle it in small socle degrees. This allows us to prove Stanley’s conjecture for all matroids of rank 3. At the end of the paper, using our method, we discuss a first possible approach to Stanley’s conjecture in full generality. Our technical work on pure O-sequences also uses very recent results of the third author and collaborators.
Similar content being viewed by others
References
Björner, A.: Homology and shellability of matroids and geometric lattices. In: White, N. (ed.) Matroid Applications, pp. 226–283. Cambridge University Press, Cambridge (1992)
Boij, M. et al.: On the shape of a pure O-sequence. Mem. Amer. Math. Soc. 218(1024), (2012)
Brown J.I., Colbourn C.J.: Roots of the reliability polynomial. SIAM J. Discrete Math. 5(4), 571–585 (1992)
Bruns W., Herzog J.: Cohen-Macaulay Rings. Cambridge Univ. Press, Cambridge (1993)
Chari M.K.: Matroid inequalities. Discrete Math. 147, 283–286 (1995)
Chari M.K.: Two decompositions in topological combinatorics with applications to matroid complexes. Trans. Amer. Math. Soc. 349(10), 3925–3943 (1997)
Constantinescu, A., Varbaro, M.: h-vectors of matroid complexes. Preprint (2012)
De Loera, J., Kemper, Y., Klee, S.: h-vectors of small matroid complexes. Electron. J. Combin. 19, #P14 (2012)
Geramita, A.V.: Inverse systems of fat points: waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals. In: The Curves Seminar at Queen’s, Vol. X, pp. 2–114. Queen’s University, Kingston, ON (1996)
Hausel T.: Quaternionic geometry of matroids. Cent. Eur. J. Math. 3(1), 26–38 (2005)
Hausel T., Sturmfels B.: Toric hyperKähler varieties. Doc. Math. 7, 495–534 (2002)
Hibi T.: What can be said about pure O-sequences?. J. Combin. Theory Ser. A 50(2), 319–322 (1989)
Iarrobino A., Kanev V.: Power Sums, Gorenstein Algebras, and Determinantal Loci. Springer, Heidelberg (1999)
Macaulay F.H.S.: Some properties of enumeration in the theory of modular systems. Proc. London Math. Soc. 26, 531–555 (1927)
Merino C.: The chip firing game and matroid complexes. Discrete Math. Theor. Comput. Sci. Proc. AA, 245–255 (2001)
Merino C., Noble S.D., Ramírez-Ibáñez M., Villarroel-Flores R.: On the structure of the h-vector of a paving matroid. European J. Combin. 33(8), 1787–1799 (2012)
Miller E., Sturmfels B.: Combinatorial commutative algebra. Springer-Verlag, New York (2005)
Neel D.L., Neudauer N.A.: Matroids you have known. Math. Mag. 82(1), 26–41 (2009)
Oh, S.: Generalized permutohedra, h-vectors of cotransversal matroids and pure O-sequences. Preprint, available on the arXiv at http://arxiv.org/abs/1005.5586
Oxley, J.G.: Matroid Theory. Oxford University Press (2006)
Proudfoot, N.: On the h-vector of a matroid complex. Unpublished note (2002)
Schweig, J.: On the h-vector of a lattice path matroid. Electron. J. Combin. 17(1), #N3 (2010)
Stanley, R.: Cohen-Macaulay complexes. In: Aigner, M. (ed.) Higher Combinatorics, pp. 51–62. Reidel, Dordrecht and Boston (1977)
Stanley, R.: Combinatorics and Commutative Algebra. Birkhäuser Boston, Inc., Boston, MA (1996)
Stanley, R.: Positivity problems and conjectures in algebraic combinatorics. In: Arnold, V. et al. (eds.) Mathematics: Frontiers and Perspectives, pp. 295–319. Amer. Math. Soc., Providence, RI (2000)
Stokes, E.: The h-vectors of matroids and the arithmetic degree of squarefree strongly stable ideals. Ph.D. Thesis, University of Kentucky (2008). Available at http://archive.uky.edu/bitstream/10225/908/Erik_Stokes-dissertation.pdf
Stokes, E.: The h-vectors of 1-dimensional matroid complexes and a conjecture of Stanley. Preprint, available on the arXiv at http://arxiv.org/abs/0903.3569
Swartz, E.: g-elements of matroid complexes. J. Combin. Theory Ser. B 88, 369–375 (2003)
Swartz E.: g-elements, finite buildings, and higher Cohen-Macaulay connectivity. J. Combin. Theory Ser. A 113(7), 1305–1320 (2006)
White, N. (ed.): Theory of Matroids. Cambridge Univ. Press, Cambridge (1986)
White, N. (ed.): Matroids Applications. Cambridge Univ. Press, Cambridge (1992)
Zanello F.: Interval conjectures for level Hilbert functions. J. Algebra 321(10), 2705–2715 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hà, H.T., Stokes, E. & Zanello, F. Pure O-Sequences and Matroid h-Vectors. Ann. Comb. 17, 495–508 (2013). https://doi.org/10.1007/s00026-013-0193-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00026-013-0193-6