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On the Shape of a Pure O-sequence
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-9961-383X
Univ Notre Dame, Notre Dame, IN, USA.
Univ Barcelona, Barcelona, Spain.
Univ Kentucky, Lexington, KY USA.
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2012 (English)In: Memoirs of the American Mathematical Society, ISSN 0065-9266, Vol. 218, no 1024, 1-+ p.Article in journal (Refereed) Published
Abstract [en]

A monomial order ideal is a finite collection X of (monic) monomials such that, whenever M is an element of X and N divides M, then N is an element of 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, (h) under bar = (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: (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.

Place, publisher, year, edition, pages
2012. Vol. 218, no 1024, 1-+ p.
Keyword [en]
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 sirnplicial complex, Macaulay's inverse system
National Category
URN: urn:nbn:se:kth:diva-99067ISI: 000305504300001OAI: diva2:541558
QC 20120719Available from: 2012-07-19 Created: 2012-07-13 Last updated: 2012-07-19Bibliographically approved

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