On the Shape of a Pure $O$-Sequence: Volume 218, Issue 1024

Mats Boij

ene 2012 · American Mathematical Soc.

Acerca de este libro electrónico

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 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.