h-vectors of matroid complexes

TL;DR

This paper classifies matroids by dimension and vertex set, identifies extremal h-vectors within each class, and proves Stanley's conjecture in several cases, including low Cohen-Macaulay type.

Contribution

It introduces a classification of matroids based on their h-vectors and confirms Stanley's conjecture for specific classes of matroids.

Findings

Identified extremal matroids with minimal and maximal h-vectors in each class.

Confirmed Stanley's conjecture for Cohen-Macaulay type ≤ 3.

Established a classification framework for matroid h-vectors.

Abstract

We partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a long-standing conjecture of Stanley. As a byproduct of this theory we establish Stanley's conjecture in various cases, for example the case of Cohen-Macaulay type less than or equal to 3.