Dual matroid polytopes and internal activity of independence complexes

TL;DR

This paper introduces a geometric approach to analyze shelling orders in matroid independence complexes, providing new tools, heuristics, and software to compare and understand their properties and address longstanding conjectures.

Contribution

It establishes a novel relation between dual matroid polytopes and independence complexes, offering a systematic method to investigate shelling orders and deepen classical results.

Findings

New relation between shellability and dual matroid polytopes

Systematic comparison method for shelling orders

Software tools for experimentation with geometric ideas

Abstract

Shelling orders are a ubiquitous tool used to understand invariants of cell complexes. Significant effort has been made to develop techniques to decide when a given complex is shellable. However, empirical evidence shows that some shelling orders are better than others. In this article, we explore this phenomenon in the case of matroid independence complexes. Based on a new relation between shellability of dual matroid polytopes and independence complexes, we outline a systematic way to investigate and compare different shellings orders. We explain how our new tools recast and deepen various classical results to the language of geometry, and suggest new heuristics for addressing two old conjectures due to Simon and Stanley. Furthermore, we present freely available software which can be used to experiment with these new geometric ideas.