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Structural properties of nested set complexes

Basile Coron, Luis Ferroni, Shiyue Li

TL;DR

The paper studies nested set complexes $\mathcal{N}(\mathcal{L},\mathcal{G})$ associated to matroids with arbitrary building sets, proving vertex decomposability and convex ear decompositions, which yields strong combinatorial and topological consequences and unifies Bergman-type results. It derives a descent-based formula for the $h$-polynomial and shows the $h$-vector is strongly flawless with the $g$-vector forming an $M$-vector in the matroid case. In the braid-matroid setting, it identifies the complex of trees with $\mathcal{N}(\Pi_n,\mathcal{G}_{\min})$, proves vertex decomposability and convex ear decomposition, and connects the $h$-polynomial to the second Eulerian polynomial via a bijection with Stirling permutations. The work also highlights open questions on unimodality, real-rootedness, and PS ear decompositions, proposing conjectures and directions for further research.

Abstract

We study structural and topological properties of nested set complexes of matroids with arbitrary building sets, proving that these complexes are vertex decomposable and admit convex ear decompositions. These results unify and generalize several recent and classical theorems on Bergman complexes and augmented Bergman complexes of matroids. As a first application, we show that the $h$-vector of a nested set complex is strongly flawless and, in particular, top-heavy. We then specialize to the boundary complex of the Deligne--Mumford--Knudsen moduli space $\overline{\mathcal{M}}_{0, n}$ of rational stable marked curves, which coincides with the complex of trees, establishing new structural decomposition theorems and deriving combinatorial formulas for its face enumeration polynomials.

Structural properties of nested set complexes

TL;DR

The paper studies nested set complexes associated to matroids with arbitrary building sets, proving vertex decomposability and convex ear decompositions, which yields strong combinatorial and topological consequences and unifies Bergman-type results. It derives a descent-based formula for the -polynomial and shows the -vector is strongly flawless with the -vector forming an -vector in the matroid case. In the braid-matroid setting, it identifies the complex of trees with , proves vertex decomposability and convex ear decomposition, and connects the -polynomial to the second Eulerian polynomial via a bijection with Stirling permutations. The work also highlights open questions on unimodality, real-rootedness, and PS ear decompositions, proposing conjectures and directions for further research.

Abstract

We study structural and topological properties of nested set complexes of matroids with arbitrary building sets, proving that these complexes are vertex decomposable and admit convex ear decompositions. These results unify and generalize several recent and classical theorems on Bergman complexes and augmented Bergman complexes of matroids. As a first application, we show that the -vector of a nested set complex is strongly flawless and, in particular, top-heavy. We then specialize to the boundary complex of the Deligne--Mumford--Knudsen moduli space of rational stable marked curves, which coincides with the complex of trees, establishing new structural decomposition theorems and deriving combinatorial formulas for its face enumeration polynomials.
Paper Structure (19 sections, 25 theorems, 62 equations, 6 figures)

This paper contains 19 sections, 25 theorems, 62 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Background
  5. Cohen-Macaulayness and shellability
  6. Vertex decomposability
  7. Convex ear decomposition
  8. Face enumeration
  9. Admissible lattices
  10. Building sets and nested sets
  11. Vertex decomposability
  12. The vertex decomposition
  13. A formula for the h-vector of a nested set complex
  14. Convex ear decomposition
  15. Braid matroids and the minimal building set
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\mathcal{L}$ be a graded lattice and let $\mathcal{G}$ be any building set of $\mathcal{L}$. If $\mathcal{L}$ admits an injective admissible map, then the nested set complex $\mathcal{N}(\mathcal{L}, \mathcal{G})$ is vertex decomposable.

Figures (6)

  • Figure 1: Relationships among the structural, topological, and numerical properties of simplicial complexes relevant to the present paper.
  • Figure 2: Vertex decomposable simplicial complexes with shedding vertices.
  • Figure 3: Attaching two $2$-dimensional simplicial balls, or "ears" to a simplicial polytope $\Delta_1$.
  • Figure 4: Deletion of a building ideal
  • Figure 5: Successive deletions of building ideals
  • ...and 1 more figures

Theorems & Definitions (67)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: provan-billera
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • ...and 57 more