AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
On the shape of a pure $O$-sequence
About this Title
Mats Boij, Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden, Juan C. Migliore, Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, Rosa M. Miró-Roig, Facultat de Matemàtiques, Department d’Àlgebra i Geometria, Gran Via des les Corts Catalanes 585, 08007 Barcelona, Spain, Uwe Nagel, Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027 and Fabrizio Zanello, Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 218, Number 1024
ISBNs: 978-0-8218-6910-9 (print); 978-0-8218-9010-3 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00647-7
Published electronically: October 3, 2011
Keywords: Pure $O$-sequence,
Artinian algebra,
monomial algebra,
unimodality,
differentiable $O$-sequence,
level algebra,
Gorenstein algebra,
enumeration,
interval conjecture,
$g$-element,
weak Lefschetz property,
strong Lefschetz property,
matroid simplicial complex,
Macaulay’s inverse system.
MSC: Primary 13D40, 05E40, 06A07, 13E10, 13H10; Secondary 05A16, 05B35, 14M05, 13F20
Table of Contents
Chapters
- 1. Introduction
- 2. Definitions and preliminary results
- 3. Differentiability and unimodality
- 4. The Interval Conjecture for Pure $O$-sequences
- 5. Enumerating pure $O$-sequences
- 6. Monomial Artinian level algebras of type two in three variables
- 7. Failure of the WLP and the SLP
- 8. Remarks on pure $f$-vectors
- 9. Some open or open-ended problems
- A. Collection of definitions and notation
Abstract
A monomial order ideal is a finite collection $X$ of (monic) monomials such that, whenever $M\in X$ and $N$ divides $M$, then $N\in X$. Hence $X$ is a poset, where the partial order is given by divisibility. If all, say $t$, maximal monomials of $X$ have the same degree, then $X$ is pure (of type $t$).
A pure $O$-sequence is the vector, $\underline {h}=(h_0=1,h_1,...,h_e)$, counting the monomials of $X$ in each degree. Equivalently, pure $O$-sequences can be characterized as the $f$-vectors of pure multicomplexes, or, in the language of commutative algebra, as the $h$-vectors of monomial Artinian level algebras.
Pure $O$-sequences had their origin in one of the early works of Stanley’s in this area, and have since played a significant role in at least three different disciplines: the study of simplicial complexes and their $f$-vectors, the theory of level algebras, and the theory of matroids. This monograph is intended to be the first systematic study of the theory of pure $O$-sequences.
Our work, which makes an extensive use of both algebraic and combinatorial techniques, in particular includes:
- Jeaman Ahn and Yong Su Shin, Generic initial ideals and graded Artinian-level algebras not having the weak-Lefschetz property, J. Pure Appl. Algebra 210 (2007), no. 3, 855–879. MR 2324612, DOI 10.1016/j.jpaa.2006.12.003
- Yousef Alavi, Paresh J. Malde, Allen J. Schwenk, and Paul Erdős, The vertex independence sequence of a graph is not constrained, Congr. Numer. 58 (1987), 15–23. Eighteenth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, Fla., 1987). MR 944684
- David Bernstein and Anthony Iarrobino, A nonunimodal graded Gorenstein Artin algebra in codimension five, Comm. Algebra 20 (1992), no. 8, 2323–2336. MR 1172667, DOI 10.1080/00927879208824466
- A. M. Bigatti and A. V. Geramita, Level algebras, lex segments, and minimal Hilbert functions, Comm. Algebra 31 (2003), no. 3, 1427–1451. MR 1971070, DOI 10.1081/AGB-120017774
- Anders Björner, The unimodality conjecture for convex polytopes, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 187–188. MR 598684, DOI 10.1090/S0273-0979-1981-14877-1
- Anders Björner, Nonpure shellability, $f$-vectors, subspace arrangements and complexity, Formal power series and algebraic combinatorics (New Brunswick, NJ, 1994) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 24, Amer. Math. Soc., Providence, RI, 1996, pp. 25–53. MR 1363505
- Mats Boij, Graded Gorenstein Artin algebras whose Hilbert functions have a large number of valleys, Comm. Algebra 23 (1995), no. 1, 97–103. MR 1311776, DOI 10.1080/00927879508825208
- Mats Boij and Dan Laksov, Nonunimodality of graded Gorenstein Artin algebras, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1083–1092. MR 1227512, DOI 10.1090/S0002-9939-1994-1227512-2
- Mats Boij and Fabrizio Zanello, Level algebras with bad properties, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2713–2722. MR 2317944, DOI 10.1090/S0002-9939-07-08829-6
- B. Boyle, Ph.D. thesis, in progress.
- B. Boyle, J. Migliore and F. Zanello: More plane partitions enumerated by the Weak Lefschetz Property, in progress.
- Holger Brenner and Almar Kaid, Syzygy bundles on $\Bbb P^2$ and the weak Lefschetz property, Illinois J. Math. 51 (2007), no. 4, 1299–1308. MR 2417428
- Jason I. Brown and Charles J. Colbourn, Roots of the reliability polynomial, SIAM J. Discrete Math. 5 (1992), no. 4, 571–585. MR 1186825, DOI 10.1137/0405047
- R. H. Bruck and H. J. Ryser, The nonexistence of certain finite projective planes, Canad. J. Math. 1 (1949), 88–93. MR 27520, DOI 10.4153/cjm-1949-009-2
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- Manoj K. Chari, Matroid inequalities, Discrete Math. 147 (1995), no. 1-3, 283–286. MR 1364520, DOI 10.1016/0012-365X(95)00122-D
- Manoj K. Chari, Two decompositions in topological combinatorics with applications to matroid complexes, Trans. Amer. Math. Soc. 349 (1997), no. 10, 3925–3943. MR 1422892, DOI 10.1090/S0002-9947-97-01921-1
- C. Chen, A. Guo, X. Jin and G. Liu: Trivariate monomial complete intersections and plane partitions, to appear, J. Commutative Algebra.
- Young Hyun Cho and Anthony Iarrobino, Hilbert functions and level algebras, J. Algebra 241 (2001), no. 2, 745–758. MR 1843323, DOI 10.1006/jabr.2001.8787
- CoCoA team.CoCoA: a system for doing computations in commutative algebra, available at http://cocoa.dima.unige.it.
- D. Cook II and U. Nagel: The Weak Lefschetz Property, Monomial Ideals, and Lozenges, to appear, Illinois J. Math.
- D. Cook II and U. Nagel, Enumerations deciding the weak Lefschetz property, preprint. Available at the arXiv at http://www.front.math.ucdavis.edu/1105.6062.
- A.V. Geramita: Inverse Systems of Fat Points: Waring’s Problem, Secant Varieties and Veronese Varieties and Parametric Spaces of Gorenstein Ideals, Queen’s Papers in Pure and Applied Mathematics, no. 102, The Curves Seminar at Queen’s (1996), Vol. X, 3–114.
- Anthony V. Geramita, Tadahito Harima, Juan C. Migliore, and Yong Su Shin, The Hilbert function of a level algebra, Mem. Amer. Math. Soc. 186 (2007), no. 872, vi+139. MR 2292384, DOI 10.1090/memo/0872
- A. V. Geramita and A. Lorenzini, Cancellation in resolutions and level algebras, Comm. Algebra 33 (2005), no. 1, 133–158. MR 2128157, DOI 10.1081/AGB-200040928
- Ira Gessel and Gérard Viennot, Binomial determinants, paths, and hook length formulae, Adv. in Math. 58 (1985), no. 3, 300–321. MR 815360, DOI 10.1016/0001-8708(85)90121-5
- I. Gessel and G. Viennot, Determinant, paths and plane partitions, Preprint, 1989.
- Heide Gluesing-Luerssen, Joachim Rosenthal, and Roxana Smarandache, Strongly-MDS convolutional codes, IEEE Trans. Inform. Theory 52 (2006), no. 2, 584–598. MR 2236175, DOI 10.1109/TIT.2005.862100
- Gerd Gotzmann, Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes, Math. Z. 158 (1978), no. 1, 61–70 (German). MR 480478, DOI 10.1007/BF01214566
- D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.
- Mark Green, Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann, Algebraic curves and projective geometry (Trento, 1988) Lecture Notes in Math., vol. 1389, Springer, Berlin, 1989, pp. 76–86. MR 1023391, DOI 10.1007/BFb0085925
- T. Há, E. Stokes and F. Zanello, Pure $O$-sequences and matroid $h$-vectors, preprint. Available on the arXiv at http://front.math.ucdavis.edu/1006.0325.
- Tadahito Harima, Juan C. Migliore, Uwe Nagel, and Junzo Watanabe, The weak and strong Lefschetz properties for Artinian $K$-algebras, J. Algebra 262 (2003), no. 1, 99–126. MR 1970804, DOI 10.1016/S0021-8693(03)00038-3
- Tadahito Harima and Junzo Watanabe, The finite free extension of Artinian $K$-algebras with the strong Lefschetz property, Rend. Sem. Mat. Univ. Padova 110 (2003), 119–146. MR 2033004
- Tamás Hausel, Quaternionic geometry of matroids, Cent. Eur. J. Math. 3 (2005), no. 1, 26–38. MR 2110782, DOI 10.2478/BF02475653
- Tamás Hausel and Bernd Sturmfels, Toric hyperKähler varieties, Doc. Math. 7 (2002), 495–534. MR 2015052
- J. Herzog and D. Popescu, The strong Lefschetz property and simple extensions, preprint. Available on the arXiv at http://front.math.ucdavis.edu/0506.5537.
- Takayuki Hibi, What can be said about pure $O$-sequences?, J. Combin. Theory Ser. A 50 (1989), no. 2, 319–322. MR 989204, DOI 10.1016/0097-3165(89)90025-3
- Anthony Iarrobino, Compressed algebras: Artin algebras having given socle degrees and maximal length, Trans. Amer. Math. Soc. 285 (1984), no. 1, 337–378. MR 748843, DOI 10.1090/S0002-9947-1984-0748843-4
- Anthony Iarrobino and Vassil Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721, Springer-Verlag, Berlin, 1999. Appendix C by Iarrobino and Steven L. Kleiman. MR 1735271
- Petteri Kaski and Patric R. J. Östergård, Classification algorithms for codes and designs, Algorithms and Computation in Mathematics, vol. 15, Springer-Verlag, Berlin, 2006. With 1 DVD-ROM (Windows, Macintosh and UNIX). MR 2192256
- T. P. Kirkman: On a Problem in Combinatorics, Cambridge Dublin Math. J. 2, (1847), 191–204.
- Jan O. Kleppe, Juan C. Migliore, Rosa Miró-Roig, Uwe Nagel, and Chris Peterson, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Math. Soc. 154 (2001), no. 732, viii+116. MR 1848976, DOI 10.1090/memo/0732
- C. Krattenthaler, Advanced determinant calculus, Sém. Lothar. Combin. 42 (1999), Art. B42q, 67. The Andrews Festschrift (Maratea, 1998). MR 1701596
- C. Krattenthaler, Another involution principle-free bijective proof of Stanley’s hook-content formula, J. Combin. Theory Ser. A 88 (1999), no. 1, 66–92. MR 1713492, DOI 10.1006/jcta.1999.2979
- Vadim E. Levit and Eugen Mandrescu, Independence polynomials of well-covered graphs: generic counterexamples for the unimodality conjecture, European J. Combin. 27 (2006), no. 6, 931–939. MR 2226428, DOI 10.1016/j.ejc.2005.04.007
- Jizhou Li and Fabrizio Zanello, Monomial complete intersections, the weak Lefschetz property and plane partitions, Discrete Math. 310 (2010), no. 24, 3558–3570. MR 2734737, DOI 10.1016/j.disc.2010.09.006
- Svante Linusson, The number of $M$-sequences and $f$-vectors, Combinatorica 19 (1999), no. 2, 255–266. MR 1723043, DOI 10.1007/s004930050055
- F.H.S. Macaulay: Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531–555.
- Percy A. MacMahon, Combinatory analysis, Chelsea Publishing Co., New York, 1960. Two volumes (bound as one). MR 0141605
- C. Merino, S.D. Noble, M. Ramírez and R. Villarroel: On the structure of the $h$-vector of a paving matroid, preprint. Available on the arXiv at http://front.math.ucdavis.edu/1008.2031.
- T. S. Michael and William N. Traves, Independence sequences of well-covered graphs: non-unimodality and the Roller-Coaster Conjecture, Graphs Combin. 19 (2003), no. 3, 403–411. MR 2007902, DOI 10.1007/s00373-002-0515-7
- Juan C. Migliore, The geometry of the weak Lefschetz property and level sets of points, Canad. J. Math. 60 (2008), no. 2, 391–411. MR 2398755, DOI 10.4153/CJM-2008-019-2
- Juan C. Migliore, Rosa M. Miró-Roig, and Uwe Nagel, Monomial ideals, almost complete intersections and the weak Lefschetz property, Trans. Amer. Math. Soc. 363 (2011), no. 1, 229–257. MR 2719680, DOI 10.1090/S0002-9947-2010-05127-X
- J. Migliore and U. Nagel, Lifting monomial ideals, Comm. Algebra 28 (2000), no. 12, 5679–5701. Special issue in honor of Robin Hartshorne. MR 1808596, DOI 10.1080/00927870008827182
- J. Migliore and U. Nagel, Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers, Adv. Math. 180 (2003), no. 1, 1–63. MR 2019214, DOI 10.1016/S0001-8708(02)00079-8
- Juan Migliore, Uwe Nagel, and Fabrizio Zanello, On the degree two entry of a Gorenstein $h$-vector and a conjecture of Stanley, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2755–2762. MR 2399039, DOI 10.1090/S0002-9939-08-09456-2
- Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
- Uwe Nagel and Tim Römer, Glicci simplicial complexes, J. Pure Appl. Algebra 212 (2008), no. 10, 2250–2258. MR 2426505, DOI 10.1016/j.jpaa.2008.03.005
- David L. Neel and Nancy Ann Neudauer, Matroids you have known, Math. Mag. 82 (2009), no. 1, 26–41. MR 2488365, DOI 10.1080/0025570x.2009.11953589
- S. Oh: Generalized permutohedra, $h$-vectors of cotransversal matroids and pure $O$-sequences, preprint. Available on the arXiv at http://front.math.ucdavis.edu/1005.5586.
- Les Reid, Leslie G. Roberts, and Moshe Roitman, On complete intersections and their Hilbert functions, Canad. Math. Bull. 34 (1991), no. 4, 525–535. MR 1136655, DOI 10.4153/CMB-1991-083-9
- Jay Schweig, On the $h$-vector of a lattice path matroid, Electron. J. Combin. 17 (2010), no. 1, Note 3, 6. MR 2578897
- Richard P. Stanley, Cohen-Macaulay complexes, 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. MR 0572989
- Richard P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57–83. MR 485835, DOI 10.1016/0001-8708(78)90045-2
- Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
- Richard P. Stanley, Positivity problems and conjectures in algebraic combinatorics, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295–319. MR 1754784
- R. Stanley: \lq\lq Enumerative Combinatorics”, Vol. I, Second Ed., Cambridge Univ. Press, Cambridge, to appear.
- Richard P. Stanley, The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236–238. MR 563925, DOI 10.1016/0001-8708(80)90050-X
- Richard P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 168–184. MR 578321, DOI 10.1137/0601021
- E. Stokes: “The $h$-vectors of matroids and the arithmetic degree of squarefree strongly stable ideals”, Ph.D. Thesis, University of Kentucky (2008).
- E. Stokes: The $h$-vectors of $1$-dimensional matroid complexes and a conjecture of Stanley, preprint. Available on the arXiv at http://front.math.ucdavis.edu/0903.3569.
- Ed Swartz, $g$-elements of matroid complexes, J. Combin. Theory Ser. B 88 (2003), no. 2, 369–375. MR 1983365, DOI 10.1016/S0095-8956(03)00038-8
- Junzo Watanabe, The Dilworth number of Artinian rings and finite posets with rank function, Commutative algebra and combinatorics (Kyoto, 1985) Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 303–312. MR 951211, DOI 10.2969/aspm/01110303
- Arthur Jay Weiss, Some new non-unimodal level algebras, ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)–Tufts University. MR 2709690
- Neil White (ed.), Theory of matroids, Encyclopedia of Mathematics and its Applications, vol. 26, Cambridge University Press, Cambridge, 1986. MR 849389
- Neil White (ed.), Matroid applications, Encyclopedia of Mathematics and its Applications, vol. 40, Cambridge University Press, Cambridge, 1992. MR 1165537
- Fabrizio Zanello, H-vectors and socle-vectors of graded artinian algebras, ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–Queen’s University (Canada). MR 2706950
- Fabrizio Zanello, A non-unimodal codimension 3 level $h$-vector, J. Algebra 305 (2006), no. 2, 949–956. MR 2266862, DOI 10.1016/j.jalgebra.2006.07.009
- Fabrizio Zanello, Interval conjectures for level Hilbert functions, J. Algebra 321 (2009), no. 10, 2705–2715. MR 2512622, DOI 10.1016/j.jalgebra.2007.09.030
- Fabrizio Zanello, Level algebras of type 2, Comm. Algebra 34 (2006), no. 2, 691–714. MR 2211949, DOI 10.1080/00927870500387986
[(i)] A characterization of the first half of a pure $O$-sequence, which yields the exact converse to a $g$-theorem of Hausel;
[(ii)] A study of (the failing of) the unimodality property;
[(iii)] The problem of enumerating pure $O$-sequences, including a proof that almost all $O$-sequences are pure, a natural bijection between integer partitions and type 1 pure $O$-sequences, and the asymptotic enumeration of socle degree 3 pure $O$-sequences of type $t$;
[(iv)] A study of the Interval Conjecture for Pure $O$-sequences (ICP), which represents perhaps the strongest possible structural result short of an (impossible?) full characterization;
[(v)] A pithy connection of the ICP with Stanley’s conjecture on the $h$-vectors of matroid complexes;
[(vi)] A more specific study of pure $O$-sequences of type 2, including a proof of the Weak Lefschetz Property in codimension 3 over a field of characteristic zero. As an immediate corollary, pure $O$-sequences of codimension 3 and type 2 are unimodal (over an arbitrary field).
[(vii)] An analysis, from a commutative algebra viewpoint, of the extent to which the Weak and Strong Lefschetz Properties can fail for monomial algebras.
[(viii)] Some observations about pure $f$-vectors, an important special case of pure $O$-sequences.