g-elements, finite buildings and higher Cohen-Macaulay connectivity

The main result is a proof that the g-vector of a simplicial complex with a convex ear decomposition is an M-vector. This is a generalization of similar results for matroid complexes. We also show that a finite building has a convex ear decomposition. This leads to connections between higher Cohen-Macaulay connectivity and increasing h-vectors.Comment: To appear in JCT A. 20 page

Poset Convex-Ear Decompositions and Applications to the Flag h-Vector

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