A lexicographic shellability characterization of geometric lattices

TL;DR

This paper characterizes geometric lattices as finite atomic lattices where every atom or join-irreducible ordering induces a lexicographic shelling, providing a new perspective on their structure.

Contribution

It introduces a novel characterization of geometric lattices based on lexicographic shellings induced by all possible orderings of atoms or join-irreducibles.

Findings

Geometric lattices are exactly those where every atom ordering induces a lexicographic shelling.

Equivalent characterization using all orderings on join-irreducibles.

Connects to McNamara's characterization of supersolvable lattices.

Abstract

Geometric lattices are characterized in this paper as those finite, atomic lattices such that every atom ordering induces a lexicographic shelling given by an edge labeling known as a minimal labeling. Equivalently, geometric lattices are shown to be exactly those finite lattices such that every ordering on the join-irreducibles induces a lexicographic shelling. This new characterization fits into a similar paradigm as McNamara's characterization of supersolvable lattices as those lattices admitting a different type of lexicographic shelling, namely one in which each maximal chain is labeled with a permutation of {1,...,n}.