Quasi-matroidal classes of ordered simplicial complexes

TL;DR

This paper introduces quasi-matroidal classes of ordered simplicial complexes, bridging matroid theory and shifted complexes, enabling new approaches to classical conjectures and extending matroid invariants.

Contribution

It defines quasi-matroidal classes that approximate matroids, allowing for inductive techniques and extensions of matroid invariants like the Tutte polynomial to shifted complexes.

Findings

Quasi-matroidal classes include pure shifted complexes and matroid independence complexes.

Extended Stanley's conjecture on $h$-vectors to these classes, verified up to rank 4.

Provided a framework for inductive methods not available in pure matroid theory.

Abstract

We introduce the notion of a quasi-matroidal class of ordered simplicial complexes: an approximation to the idea of a matroid cryptomorphism in the landscape of ordered simplicial complexes. A quasi-matroidal class contains pure shifted simplicial complexes and ordered matroid independence complexes. The essential property is that if a fixed simplicial complex belongs to this class for every ordering of its vertex set, then it is a matroid independence complex. Some examples of such classes appear implicitly in the matroid theory literature. We introduce various such classes that highlight different apsects of matroid theory and its similarities with the theory of shifted simplicial complexes. For example, we lift the study of objects like the Tutte polynomial and nbc complexes to a quasi-matroidal class that allows us to define such objects for shifted complexes. Furthermore, some of the quasi-matroidal classes are amenable to inductive techniques that can't be applied directly in the context of matroid theory. As an example, we provide a suitable setting to reformulate and extend conjecture of Stanley about $h$-vectors of matroids which is expected to be tractable with techniques that are out of reach for matroids alone. This new conjecture holds for pure shifted simplicial complexes and matroids of rank up to 4.