Two Decompositions in Topological Combinatorics with Applications to Matroid Complexes

This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of M-shellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the rank-numbers of M-shellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex ear-decomposition, and, using results of Kalai and Stanley on $h$-vectors of simplicial polytopes, we show that $h$-vectors of pure rank-$d$ simplicial complexes that have this property satisfy $h_{0} \leq h_{1} \leq \cdots \leq h_{[d/2]}$ and $h_{i} \leq h_{d-i}$ for $0 \leq i \leq [d/2]$. We then show that the abstract simplicial complex formed by the collection of independent sets of a matroid (or matroid complex) admits a special type of convex ear-decomposition called a PS ear-decomposition. This enables us to construct an associated M-shellable poset, whose set of rank-numbers is the $h$-vector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi that the $h$-vector of a matroid complex satisfies the above two sets of inequalities.