Convex-Ear Decompositions and the Flag h-Vector

TL;DR

This paper introduces a theorem for convex-ear decompositions of certain posets, leading to new inequalities for h-vectors and flag h-vectors in geometric and Cohen-Macaulay complexes.

Contribution

It presents a novel theorem enabling convex-ear decompositions for specific posets, with applications to geometric lattices and Cohen-Macaulay complexes.

Findings

Derived new inequalities for h-vectors of shellable complexes

Established inequalities for flag h-vectors of Cohen-Macaulay complexes

Applied convex-ear decompositions to geometric lattices

Abstract

We prove a theorem allowing us to find convex-ear decompositions for rank-selected subposets of posets that are unions of Boolean sublattices in a coherent fashion. We then apply this theorem to geometric lattices and face posets of shellable complexes, obtaining new inequalities for their h-vectors. Finally, we use the latter decomposition to prove new inequalities for the flag h-vectors of face posets of Cohen-Macaulay complexes.