## Rocky Mountain Journal of Mathematics

### The unimodality of pure $O$-sequences of type two in four variables

#### Abstract

Since the 1970's, great interest has been taken in the study of pure $O$-sequences, which, due to Macaulay's theory of inverse systems, have a bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these according to their shape. Macaulay's theorem immediately gives us that all Artinian algebras in two variables have unimodal Hilbert functions. Furthermore, it has been shown that all Artinian level monomial algebras of type two in three variables have unimodal Hilbert functions. This paper will classify all Artinian level monomial algebras of type two in four variables into two classes of ideals, prove that they are strictly unimodal and show that one of the classes is licci.

#### Article information

Source
Rocky Mountain J. Math., Volume 45, Number 6 (2015), 1781-1799.

Dates
First available in Project Euclid: 14 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1457960334

Digital Object Identifier
doi:10.1216/RMJ-2015-45-6-1781

Mathematical Reviews number (MathSciNet)
MR3473154

Zentralblatt MATH identifier
1332.05157

#### Citation

Boyle, Bernadette. The unimodality of pure $O$-sequences of type two in four variables. Rocky Mountain J. Math. 45 (2015), no. 6, 1781--1799. doi:10.1216/RMJ-2015-45-6-1781. https://projecteuclid.org/euclid.rmjm/1457960334

#### References

• M. Boij, J. Migliore, R. Miró-Roig, U. Nagel and F. Zanello, On the shape of a pure $O$-sequence, Mem. Amer. Math. Soc. 218 (2012), no. 2024.
• H. Brenner and A. Kaid, Syzygy Bundles on $\mathbb{P}$ and the weak Lefschetz Ppoperty, Illinois J. Math. 51 (2007), 1299–1308.
• M.K. Chari, Two decompositions in topological combinatorics with applications to matroid complexes, Trans. Amer. Math. Soc. 349 (1997), 3925–3943.
• 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 Pure Appl. Math. 102, 3–114.
• A.V. Geramita, T. Harima, J. Migliore and Y.S. Shin, The Hilbert function of a level algebra, Mem. Amer. Math. Soc. 186 (2007), No. 872.
• T. Harima, J. Migliore, U. Nagel and J. Watanabe, The weak and strong Lefschetz properties for Artinian $K$-algebras, J. Alg. 262 (2003), 99–126.
• T. Hausel, Quaternionic geometry of matroids, Cent. Europ. J. Math. 3 (2005), 26–38.
• J. Herzog and T. Hibi, Monomial ideals, Springer-Verlag, London 2011.
• T. Hibi, What can be said about pure $O$-sequences?, J. Comb. Theor. 50 (1989), 319–322.
• C. Huneke, and B. Ulrich, Liaison of monomial ideals, Bull. Lond. Math. Soc. 39 (2007), 384–392.
• A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lect. Notes Math. 1721, Springer-Verlag, Berlin, 1999.
• P. Kaski, P. Östergård and R.J. Patric, Classification algorithms for codes and designs, Algor. Comp. Math. 15, Springer-Verlag, Berlin, 2006.
• J. Kleppe, R. Miró-Roig, J. Migliore, U. Nagel and C. Peterson, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Soc. 154 (2001), No. 732.
• C. Merino, S.D. Noble, M. Ramírez-Ibáñez and R. Villarroel-Flores, On the structure of the $h$-vector of a paving matroid, Europ. J. Comb. 33 (2012), 1787–1799.
• J. Migliore, Introduction to liaison theory and deficiency modules, Birkhäuser Boston, Progr. Math. 165, 1998.
• S. Oh, Generalized permutohedra, $h$-vectors of cotransversal matroids and pure $O$-sequences, preprint. Available on the arXiv at http://arxiv.org/abs/1005.5586.
• L. Reid, L. Roberts and M. Roitman, On complete intersections and their Hilbert functions, Canad. Math. Bull. 34 (1991), 525–535.
• J. Schweig, On the $h$-vector of a lattice path matroid, Electr. J. Comb. 17 (2010), N3.
• R. Stanley, Cohen-Macaulay complexes, in Higher combinatorics, M. Aigner, ed., Reidel, Dordrecht, 1977.
• R. Stanley, Weyl groups, The hard Lefschetz theorem, and the Sperner property, SIAM J. Alg. Discr. Meth. 1 (1980), 168–184.
• E. Stokes, 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.
• J. Watanabe, The Dilworth number of Artinian rings and finite posets with rank function, Comm. Alg. Combin. Adv. Stud. Pure Math. 11, Kinokuniya Co. North Holland, Amsterdam, 1987.
• F. Zanello, A non-unimodal codimesion three level $h$-vector, J. Alg. 305 (2006), 949–956.