Nested Set Complexes in Matroid Theory

• The topic is defined as nested set complexes, which are simplicial complexes built from building sets of matroid lattices, bridging the order complex and the Bergman complex.
• They are geometrically realized by mapping flats to incidence vectors, yielding spherical and polytopal models that elucidate matroid subdivisions and direct sum decompositions.
• Their structural properties, including purity, shellability, and vertex decomposability, support advanced enumerative invariants and applications in moduli spaces and phylogenetic trees.

A nested set complex is a simplicial complex associated to a matroid (or oriented matroid) built from the combinatorics of the lattice of flats via the notion of a building set. These complexes coarsely interpolate between the order complex of the proper part of the lattice of flats and the Bergman complex, giving polyhedral and topological insights into matroid subdivisions, matroid types, and their direct sum decompositions. Nested set complexes have deep relationships with the geometry of matroid polytopes, realization as subcomplexes of nestohedra, connections to shellability and vertex decomposability, and applications in the theory of moduli spaces, oriented matroid polytopes, and subspace arrangements.

1. Building Sets and Nested Set Complexes

Let $M$ be a (loopless) matroid of rank $r$ on ground set $E=\{1,\dotsc, n\}$, with geometric lattice of flats $\mathcal{L}_M$, bottom element $\hat{0}=\emptyset$, and top $\hat{1}=E$. A building set $$\mathcal{G}\subseteq \mathcal{L}_M\setminus\{\hat{0}\}$$ is defined by the property that for every flat $X>\hat{0}$, the interval $[\hat{0}, X]$ factors as a product:

$[\hat{0}, X]\cong \prod_{i=1}^k [\hat{0}, G_i]$

where $G_1,\dotsc, G_k$ are the maximal elements of $\{G\in \mathcal{G}: G\leq X\}$.

Typical choices:

• Maximal: $$\mathcal{G}_{\max} = \mathcal{L}_M\setminus\{\hat{0}\}$$,
• Minimal: $$\mathcal{G}_{\min} = \{\text{nontrivial irreducible (connected) flats}\} \cup \{E\}$$.

A subset $S\subseteq \mathcal{G}$ is nested if for every set of pairwise incomparable $G_1, \dotsc, G_k\in S$ ($k\geq 2$), the join $G_1\vee\dotsb\vee G_k\notin \mathcal{G}$. The collection of nested sets forms the nested set complex $\mathcal{N}(\mathcal{L}_M, \mathcal{G})$, a simplicial complex on the vertex set $\mathcal{G}$.

For $\mathcal{G}_{\max}$, this complex is the order complex of the proper part of $\mathcal{L}_M$; for $\mathcal{G}_{\min}$, it gives the minimal, coarsest subdivision. These combinatorial structures are observed in the lattice of flats, but variants are definable via the Las Vergnas face lattice for oriented matroids (Dlugosch, 2011, Coron et al., 8 Jan 2026, Mantovani et al., 19 Sep 2025).

2. Polyhedral and Topological Realizations

Nested set complexes possess canonical geometric realizations. Each $G\in\mathcal{G}$ is mapped to its incidence vector $e_G=\sum_{i\in G} e_i$ in $\mathbb{R}^n$. Cones generated by $e_G$ for all $G\in S$ with $S$ nested yield a fan invariant under translations along $\mathbb{R}(1,\dotsc, 1)$; intersecting this fan with $\sum_i \omega_i=0$ and the unit sphere gives a spherical realization of $\mathcal{N}(\mathcal{L}_M,\mathcal{G})$ (Dlugosch, 2011).

If $M$ is an oriented matroid realizing a vector configuration $A$, then the facial nested complex has a realization as the boundary complex of a polytope (an acyclonestohedron), obtained as a section of a nestohedron by the evaluation space of $A$ (Mantovani et al., 19 Sep 2025).

For the (maximal) building set, the geometric realization of the nested set complex yields the Bergman complex. General nested set complexes for intermediate building sets produce subdivisions of the Bergman complex, interpolating between the minimal and maximal cases (Dlugosch, 2011).

3. Subdivisions Between Order Complex and Bergman Complex

Nested set complexes sit functorially between two prominent matroidal objects:

• Order complex of $\mathcal{L}_M \setminus \{\hat{0}, \hat{1}\}$,
• Bergman complex $B(M)$.

Every choice of building set $\mathcal{G}$ with $E\in\mathcal{G}$ yields a subdivision of $B(M)$ by the realization of $\mathcal{N}(\mathcal{L}_M,\mathcal{G})$. The minimal nested set complex gives a coarse simplicial subdivision; the maximal building set gives the finest simplicial subdivision (order complex).

For any nested set face (corresponding to a nested set $\Gamma$), the associated matroid type $M_\omega$ admits a direct sum decomposition:

$$M_\omega \cong \bigoplus_{\alpha\in \Pi}\, M \left[ \left( \bigcap\Gamma_\alpha \right) \setminus \alpha,\, \bigcap\Gamma_\alpha \right]$$

where $$\Gamma_\alpha = \{F\in\Gamma:\, \alpha \subseteq F\}$$ and $\Pi$ is the partition of $E$ obtained as the coarsest partition not mixing $F$ and $E\setminus F$ for any $F\in\Gamma$. This decomposition is finest possible for faces of the Bergman complex and has substantial utility for matroid theory (Dlugosch, 2011).

Addition or removal of elements to $\mathcal{G}$ corresponds combinatorially to stellar subdivisions (blow-ups) of the complex, producing intermediate nested set subdivisions of varying granularity (Dlugosch, 2011, Mantovani et al., 19 Sep 2025).

4. Structural Properties: Purity, Shellability, and Decomposability

Nested set complexes of matroids are pure simplicial complexes of dimension $r-1$ (matroid of rank $r+1$), as all maximal nested sets have cardinality $r$ (Coron et al., 8 Jan 2026).

They are vertex decomposable and hence shellable for arbitrary building sets, a property established via inductive arguments using admissible labelings. The lexicographic order on the coordinate vectors assigned to maximal nested sets provides a concrete shelling order, yielding a new geometric proof that specializes to the classical shellability of the order complex (Backman et al., 8 Jan 2026). Consequently, all nested set complexes of matroids, including those arising from oriented matroids, are vertex decomposable, shellable, and, therefore, doubly Cohen–Macaulay (Coron et al., 8 Jan 2026, Backman et al., 8 Jan 2026).

Further, they admit convex ear decompositions in the sense of Chari, leading to strong inequalities for their $h$-vectors (e.g., top-heaviness, flawlessness, and unimodality). The $g$-vector is always an $M$-vector, with the complementary vector a sum of $M$-vectors (Coron et al., 8 Jan 2026).

This structural landscape is shared by all prominent specializations:

• Bergman complex ($\mathcal{G}_{\max}$): shellable/order complex,
• Augmented Bergman complex: shellable,
• Complex of phylogenetic trees (minimal building set for partition lattice): shellable and vertex decomposable.

5. Enumerative Invariants and Face Formulas

The $h$-vector $(h_0,\dotsc,h_r)$ of $\mathcal{N}(M,\mathcal{G})$ is determined from the face vector $(f_i)$ by

$$\sum_{i=0}^r f_i (t-1)^{r-i} = \sum_{i=0}^r h_i t^{r-i}.$$

A refined enumeration expresses $h$-polynomials in terms of descent statistics: for each admissible atom labeling, $h_i$ counts the number of maximal nested sets having $i$ descents in their tree structure (Coron et al., 8 Jan 2026). For the Bergman complex, $h$-polynomials reduce to (shifted) characteristic polynomials of the matroid. Related formulas for augmented Bergman complexes connect to the Tutte polynomial.

Closed-form $f$-vector or $h$-vector formulas are only available in special cases. In general, the descent-based generating function gives a uniform combinatorial approach (Coron et al., 8 Jan 2026).

For the complex of phylogenetic trees, the $h$-polynomial is the second Eulerian polynomial $A_n^{(2)}(t)$—a generating function over Stirling permutations—known to be real-rooted and $\gamma$-positive (Coron et al., 8 Jan 2026).

6. Realization as Subcomplexes and Embeddings

Facial nested complexes associated to oriented matroids are constructed as sequences of combinatorial blow-ups (stellar subdivisions) on the Las Vergnas face lattice, yielding new oriented matroids. If realizable, these complexes are polytopal and correspond to sections of nestohedra, such as the acyclonestohedron (Mantovani et al., 19 Sep 2025).

A general embedding theorem applies: any finite atomic lattice admits an injective atom-to-boolean representation, permitting nested set complexes to be realized as full subcomplexes of boolean nested complexes (nestohedra). There is systematic theory predicting when one nested set complex embeds into another via order- or join-preserving lattice embeddings (Mantovani et al., 19 Sep 2025). This framework unifies poset associahedra, type $B$ nestohedra, permutopermutohedra, and other families associated to both matroid and face lattices.

Applied to the inclusion of the face lattice into the flat lattice for an oriented matroid, positive Bergman complexes and their compactifications embed into the full Bergman complex, substantiating links between oriented matroid theory and classical matroid theory (Mantovani et al., 19 Sep 2025).

7. Connections and Applications

Nested set complexes are central to several areas:

• Moduli spaces: The boundary complex of $\overline{\mathcal{M}}_{0,n+1}$ is the minimal nested set complex of the partition lattice, coinciding with the complex of trees (Coron et al., 8 Jan 2026).
• Phylogenetic trees: As above, the theory yields combinatorial enumerations and topological properties.
• Wonderful compactifications: Their boundary complexes are nested set complexes arising from face lattices (Mantovani et al., 19 Sep 2025).
• Polytope theory: Nestohedra and related objects, such as acyclonestohedra, type $B$ nestohedra, and poset associahedra are constructed as polytopal models understanding subspace arrangements and their combinatorics (Mantovani et al., 19 Sep 2025).

These connections emphasize the structural utility of nested set complexes in encoding configurations, subdivisions, and decompositions relevant in algebraic geometry, combinatorial topology, and polyhedral theory.

---

Key references: (Dlugosch, 2011, Coron et al., 8 Jan 2026, Backman et al., 8 Jan 2026, Mantovani et al., 19 Sep 2025)