Convex Ear Decompositions in Complexes

• Convex ear decompositions are a method to partition pure simplicial complexes into a polytopal boundary and sequential ball-like 'ears' attached along their boundaries.
• They leverage combinatorial shellability, EL-labelings, and convex embeddings to ensure top-heavy h-vector inequalities and robust Cohen–Macaulay properties.
• These decompositions apply to matroids, geometric lattices, and nested set complexes, unifying and extending classical combinatorial and topological results.

A convex ear decomposition (c.e.d.) of a pure simplicial complex provides a powerful structural tool in topological and combinatorial analysis, yielding deep consequences for the $h$-vector, flag $h$-vector, and Cohen–Macaulay properties. These decompositions systematically partition a complex into a union of polytopal boundaries and topological balls, "ears," glued along boundaries with strict convexity and intersection properties. Originally developed in the context of face posets, geometric lattices, Bergman and augmented Bergman complexes of matroids, and later generalized to nested set complexes on arbitrary building sets, c.e.d.'s unify and extend shellability, vertex decomposability, and strong algebraic properties across broad classes of combinatorial complexes (Schweig, 2010, Athanasiadis et al., 2024, Coron et al., 8 Jan 2026).

1. Formal Definition and Construction Principles

Let $\Delta$ be a pure $(d-1)$-dimensional simplicial complex. A convex ear decomposition is a sequence of pure subcomplexes $$\Delta_1, \Delta_2, \ldots, \Delta_m \subseteq \Delta$$ such that:

• (i) $\Delta = \bigcup_{i=1}^m \Delta_i$;
• (ii) $\Delta_1$ is isomorphic to the boundary complex of a simplicial $d$-polytope;
• (iii) For each $k \geq 2$, $\Delta_k$ is homeomorphic to a $(d-1)$-ball, isomorphic to a proper subcomplex of a $d$-polytope boundary, and satisfies $$\partial \Delta_k = \Delta_k \cap (\Delta_1 \cup \cdots \cup \Delta_{k-1})$$.

Each $\Delta_k$ with $k \geq 2$ is called an "ear," and the process ensures that ears are attached only along their full boundaries, preserving topological and combinatorial regularity (Schweig, 2010, Athanasiadis et al., 2024, Coron et al., 8 Jan 2026).

2. Main Structural Theorems and Proof Outlines

The central theorem (e.g., Theorem 3.1 in (Schweig, 2010)) establishes that order complexes of rank-selected subposets with suitable Boolean or matroidal structures admit convex ear decompositions. In the context of matroids or nested set complexes, the ears are indexed by nbc-bases (or bases in lexicographic order) and constructed as polytopal boundaries or their sub-balls. The proof proceeds by induction on the ordering of subposets or bases, attaching each ear via careful EL-labelings and combinatorial shellings, ensuring boundary intersection properties hold at each step (Schweig, 2010, Athanasiadis et al., 2024, Coron et al., 8 Jan 2026).

Convexity is guaranteed by realizing each subcomplex as embedded in the boundary of a polytope, often a stellohedron or simplex, while shellability is controlled by descent-set or atom-label filtrations.

3. Classes of Complexes Admitting Convex Ear Decompositions

Convex ear decompositions have been constructed for a rich array of complexes, including:

• Order complexes of rank-selected subposets where the poset can be coherently expressed as a union of Boolean sublattices (Schweig, 2010);
• Geometric lattices (lattices of flats of simple matroids), with ears corresponding to Boolean sublattices indexed by nbc-bases (Schweig, 2010);
• Face posets of shellable complexes, using the shelling order of facets to produce Boolean sublattices associated to each simplex (Schweig, 2010);
• Bergman and augmented Bergman complexes of matroids, with ears realized as boundaries of stellohedra (matroid base polytopes), and intersection patterns guided by the ordering of bases (Athanasiadis et al., 2024, Coron et al., 8 Jan 2026);
• Nested set complexes of matroids with arbitrary building sets, extending convex ear decompositions to the full generality of nested set frameworks (Coron et al., 8 Jan 2026).

A comparative summary of key classes:

Complex Type	Ears Indexed By	Presentation of Ears
Rank-selected poset order complexes	Maximal Boolean components / nbc-bases	Boolean polytope boundaries
Bergman/augmented Bergman complexes	Matroid bases	Boundaries of stellohedra
Nested set complexes (general G)	nbc-bases	Polytopal boundaries/subcomplexes

4. Algebraic and Combinatorial Consequences

Complexes admitting a convex ear decomposition satisfy strong algebraic and combinatorial properties:

• Double Cohen–Macaulayness over any field (Athanasiadis et al., 2024);
• Top-heavy $h$-vector inequalities: $$h_0 \leq h_1 \leq \ldots \leq h_{\lfloor d/2 \rfloor}$$ and $h_i < h_{d-i}$ for $0 \leq i < d/2$ (Schweig, 2010, Athanasiadis et al., 2024, Coron et al., 8 Jan 2026);
• $g$-vector is an M-vector (the sequence of first differences of the first half of $h$-vector entries forms the degree sequence of a monomial order ideal) (Schweig, 2010, Coron et al., 8 Jan 2026);
• Strong flawlessness and additional inequalities for $h$- and flag $h$-vectors in the nested set complex case (Coron et al., 8 Jan 2026);
• New flag $h$-vector inequalities for Cohen–Macaulay simplicial complexes, obtained by dominance relations between descent-set classes (Schweig, 2010).

A plausible implication is that, while convex ear decompositions induce top-heavy $h$-vectors and Cohen–Macaulayness, they do not guarantee log-concavity or unimodality of $f$- or $h$-vectors; explicit counterexamples are constructed in the context of large uniform matroids (Athanasiadis et al., 2024).

5. Decomposition Methodologies and Examples

For matroidal or poset-derived complexes, the procedure for forming a c.e.d. typically follows:

• Fix a total order on atoms/ground set.
• Identify nbc-bases or bases, order them lexicographically.
• For each base, form a subcomplex (e.g., stellohedral sphere or simplex boundary) whose faces correspond to chains or nested sets compatible with the base.
• At each step, select, filter, and attach new ears along their boundaries, ensuring intersection conditions.
• Verify ball and convexity properties using polytopal embeddings.

An explicit example: For the uniform rank-2 matroid $U_{2,3}$, the augmented Bergman complex decomposes into three ears (hexagons), each glued along contiguous segments, with the ears corresponding to the three bases $\{1,2\},\{1,3\},\{2,3\}$ (Athanasiadis et al., 2024). For the nested set complex of $U_{3,4}$, one obtains a cyclic "big" 3-cycle with three path-like ears attached, each corresponding to nbc-bases (Coron et al., 8 Jan 2026).

6. Extensions and Generalizations

Convex ear decompositions encompass and generalize shellability, vertex decomposability, and PS-ear decompositions in several ways. The framework in (Coron et al., 8 Jan 2026) demonstrates that the methodology applies uniformly across all nested set complexes of matroids with arbitrary building sets, thus subsuming constructions for ordinary/augmented Bergman complexes and even the complex of trees as the braid arrangement case. The approach leverages the combinatorics of nbc-bases, EL-labelings, and building set restrictions, revealing a unified theory with broad algebraic and topological ramifications.

7. Limitations and Notable Phenomena

While c.e.d.-admitting complexes showcase top-heavy $h$-vectors and strong Cohen–Macaulayness, they do not exhibit universal unimodality or log-concavity for $f$- and $h$-vectors. For instance, the augmented Bergman complex of $U_{4,189}$ has an $f$-polynomial failing log-concavity, and the $h$-vector of $U_{5,8}$ is non-unimodal, highlighting a boundary to these structural decompositions' combinatorial implications.

The c.e.d. technique thus refines but does not supplant other deep combinatorial conjectures, and the sharpness of $h$-vector inequalities under c.e.d. remains a locus of ongoing research (Athanasiadis et al., 2024).