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

Christos A. Athanasiadis, Luis Ferroni

TL;DR

This paper studies the face enumeration of augmented Bergman complexes $\Delta_{\mathsf{M}}$ of matroids, proving they admit a convex ear decomposition. This structural result implies that $\Delta_{\mathsf{M}}$ is doubly Cohen–Macaulay and that its $h$-vector is top-heavy, while yielding explicit convolution formulas for $f$- and $h$-polynomials. The authors also construct counterexamples showing that top-heaviness does not guarantee unimodality or log-concavity in general, and they draw a novel connection to augmented Chow polynomials by expressing $h(\Delta_{\mathsf{U}_{d,n}},x)$ in terms of augmented Chow ring Hilbert–Poincaré series. Together, these results deepen understanding of the combinatorial and algebraic structure of matroid-associated complexes and their connections to Chow theory.

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.

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

TL;DR

This paper studies the face enumeration of augmented Bergman complexes of matroids, proving they admit a convex ear decomposition. This structural result implies that is doubly Cohen–Macaulay and that its -vector is top-heavy, while yielding explicit convolution formulas for - and -polynomials. The authors also construct counterexamples showing that top-heaviness does not guarantee unimodality or log-concavity in general, and they draw a novel connection to augmented Chow polynomials by expressing in terms of augmented Chow ring Hilbert–Poincaré series. Together, these results deepen understanding of the combinatorial and algebraic structure of matroid-associated complexes and their connections to Chow theory.

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 -vectors. We provide some formulas for computing the -polynomials of these complexes and exhibit examples which show that, in general, they are neither log-concave nor unimodal.

Paper Structure

This paper contains 9 sections, 7 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Simplicial complexes and matroids
  3. Face enumeration
  4. Shellability
  5. Convex ear decompositions
  6. Matroids
  7. Proof of Theorem \ref{['thm:main']}
  8. On unimodality and log-concavity
  9. A connection to augmented Chow polynomials

Key Result

Theorem 1.2

The augmented Bergman complex $\Delta_{{\mathsf{M}}}$ has a convex ear decomposition for every matroid ${\mathsf{M}}$. In particular, $\Delta_{{\mathsf{M}}}$ is doubly Cohen--Macaulay, and for $0 \le i \le d/2$, where $d$ is the rank of ${\mathsf{M}}$ and $(h_i(\Delta_{{\mathsf{M}}}))_{0 \le i \le d}$ is the $h$-vector of $\Delta_{{\mathsf{M}}}$.

Figures (1)

  • Figure 1: The simplicial complex of Example \ref{['ex:one-dim']}.

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem 1.2
  • Remark 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • Example 2.5
  • Remark 2.6
  • Proposition 2.7
  • ...and 14 more