On the shape of a pure O-sequence

By M. Boij, Juan Migliore, R. Mir'o-Roig, Uwe Nagel, Fabrizio Zanello. 2010

70 citations recorded by the source corpus.

An order ideal is a finite poset X of (monic) monomials such that, whenever M is in X and N divides M, then N is in X. 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, h=(1,h_1,...,h_e), counting the monomials of X in each degree. Equivalently, in the language of commutative algebra, pure O-sequences are the h-vectors of monomial Artinian level algebras. Pure O-sequences had their origin in one of Richard Stanley's early works in this area, and have since played a significant role in at least three disciplines: the study of simplicial complexes and their f-vectors, level algebras, and matroids. This monograph is intended to be the first systematic study of the theory of pure O-sequences. Our work, making an extensive use of algebraic and combinatorial techniques, includes: (i) A characterization of the first half of a pure O-sequence, which gives the exact converse to an algebraic 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, and the asymptotic enumeration of socle degree 3 pure O-sequences of type t; (iv) The Interval Conjecture for Pure O-sequences (ICP), which represents perhaps the strongest possible structural result short of an (impossible?) characterization; (v) A pithy connection of the ICP with Stanley's matroid h-vector conjecture; (vi) A specific study of pure O-sequences of type 2, including a proof of the Weak Lefschetz Property in codimension 3 in characteristic zero. As a corollary, pure O-sequences of codimension 3 and type 2 are unimodal (over any field); (vii) An analysis 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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/17511010ad52e621923746f2bd47449c00b24fb5

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In Algebra and Number Theory

  1. Uwe Nagel
  2. Juan Migliore
  3. R. Mir'o-Roig
  4. A. V. Geramita
  5. Karim A. Adiprasito
  6. Brian Harbourne

In Computational Theory and Mathematics

  1. Uwe Nagel
  2. Juan Migliore
  3. R. Mir'o-Roig
  4. A. V. Geramita
  5. Karim A. Adiprasito

In Discrete Mathematics and Combinatorics

  1. Juan Migliore
  2. Uwe Nagel
  3. R. Mir'o-Roig

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