Jay Schweig
Jay Schweig

Ph.D. (2008) Cornell University

First Position
Dissertation
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Abstract: Possibly the most fundamental combinatorial invariant associated to a finite simplicial complex is its fvector, the integral sequence expressing the number of faces of the complex in each dimension. The hvector of a complex is obtained by applying a simple invertible transformation to its fvector, and thus the two contain the same information. Because some properties of the fvector are more easily expressed after applying this transformation, the hvector has been the subject of much study in geometric and algebraic combinatorics. A convexear decomposition, first introduced by Chari, is a way of writing a simplicial complex as a union of subcomplexes of simplicial polytope boundaries. When a (d – 1)dimensional complex admits such a decomposition, its hvector satisfies, for i < d/2, h_{i} ≤ h_{i + 1} and h_{i} ≤ h_{d – i}. Furthermore, its gvector is an Mvector.
We give convexear decompositions for the order complexes of rankselected subposets of supersolvable lattices with nowherezero Möbius functions, rankselected subposets of geometric lattices, and rankselected face posets of shellable complexes (when the rankselection does not include the maximal rank). Using these decompositions, we are able to show inequalities for the flag hvectors of supersolvable lattices and face posets of CohenMacaulay complexes.
Finally, we turn our attention to the hvectors of lattice path matroids. A lattice path matroid is a certain type of transversal matroid whose bases correspond to planar lattice paths. We verify a conjecture of Stanley in the special case of lattice path matroids and, in doing so, introduce an interesting new class of monomial order ideals.