Poset Convex-Ear Decompositions and Applications to the Flag h-Vector
| dc.contributor.author | Schweig, Jay | |
| dc.date.accessioned | 2008-04-21T19:54:00Z | |
| dc.date.available | 2008-04-21T19:54:00Z | |
| dc.date.issued | 2008-04-21T19:54:00Z | |
| dc.description.abstract | Possibly the most fundamental combinatorial invariant associated to a finite simplicial complex is its f-vector, the integral sequence expressing the number of faces of the complex in each dimension. The h-vector of a complex is obtained by applying a simple invertible transformation to its f-vector, and thus the two contain the same information. Because some properties of the f-vector are easier expressed after applying this transformation, the h-vector has been the subject of much study in geometric and algebraic combinatorics. A convex-ear 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 h-vector satisfies, for $i < d/2$, $h_i \leq h_{i+1}$ and $h_i \leq h_{d-i}$. Furthermore, its g-vector is an M-vector. We give convex-ear decompositions for the order complexes of rank-selected subposets of supersolvable lattices with nowhere-zero M\"obius functions, rank-selected subposets of geometric lattices, and rank-selected face posets of shellable complexes (when the rank-selection does not include the maximal rank). Using these decompositions, we are able to show inequalities for the flag h-vectors of supersolvable lattices and face posets of Cohen-Macaulay complexes. Finally, we turn our attention to the h-vectors 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. | en_US |
| dc.identifier.other | bibid: 6397051 | |
| dc.identifier.uri | https://hdl.handle.net/1813/10735 | |
| dc.language.iso | en_US | en_US |
| dc.subject | Mathematics | en_US |
| dc.subject | Combinatorics | en_US |
| dc.subject | Partially ordered sets | en_US |
| dc.subject | h-vector | en_US |
| dc.title | Poset Convex-Ear Decompositions and Applications to the Flag h-Vector | en_US |
| dc.type | dissertation or thesis | en_US |
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