A convex ear decomposition of the augmented Bergman complex of a matroid

TL;DR

This paper proves that augmented Bergman complexes of matroids have a convex ear decomposition, leading to properties like being doubly Cohen–Macaulay and having top-heavy h-vectors, with formulas for their h-polynomials.

Contribution

It introduces a convex ear decomposition for augmented Bergman complexes, establishing new topological and combinatorial properties for these complexes.

Findings

Augmented Bergman complexes are doubly Cohen–Macaulay.

They have top-heavy h-vectors.

Examples show they are not always log-concave or unimodal.

Abstract

In recent work of Braden, Huh, Matherne, Proudfoot and Wang, a class of simplicial complexes associated to matroids, called augmented Bergman complexes, was introduced. The present article concerns the face enumeration of these complexes. We prove that the augmented Bergman complex of any matroid admits a convex ear decomposition and deduce that augmented Bergman complexes are doubly Cohen--Macaulay and that they have top-heavy $h$-vectors. We provide some formulas for computing the $h$-polynomials of these complexes and exhibit examples which show that, in general, they are neither log-concave nor unimodal.