Polytope of Unlabeled Matroids

• Polytope of unlabeled matroids is a convex-geometric model representing isomorphism classes of matroids on a fixed ground set via unique Schubert expansions.
• It systematically characterizes extremal matroids by alternating maximization and minimization of valuative invariants, linking combinatorial optimization to polyhedral geometry.
• Explicit constructions in ranks 2 and 3 illustrate the polytope’s structure, revealing its facial organization and applications in testing representability.

Searching arXiv for the cited papers to ground the article in current literature. {"query":"arXiv (Bonin, 19 Jul 2025) Characterizations of certain matroids by maximizing valuative invariants"} {"query":"arXiv (Collins et al., 12 May 2026) The polytope of all matroids in ranks 2 and 3"} The polytope of all unlabeled matroids is a convex-geometric model for isomorphism classes of matroids of fixed rank and fixed ground-set size. For integers $0 \le r \le n$, one associates to each rank-$r$ matroid on an $n$-element ground set a coordinate vector coming from its unique Schubert-expansion, or equivalently from the Derksen–Fink decomposition of the symmetrized indicator function of its base polytope, and takes the convex hull of all such vectors. In the notation of Bonin, this polytope is $\Omega_{n,r}$; in the notation of Collins and Schleis, it is $\Omega_{r,n}$. Its vertices are called extremal matroids, and its faces encode optimization problems for valuative invariants. This polytope was introduced by Ferroni and Fink and has subsequently been developed through general structural results, new infinite families of extremal matroids, and explicit constructions in ranks $2$ and $3$ (Ferroni et al., 27 Feb 2025, Bonin, 19 Jul 2025, Collins et al., 12 May 2026).

1. Definition and coordinate models

Fix integers $0 \le r \le n$. One formulation begins with the set of isomorphism types of nested matroids of rank $r$ on $[n]$, say $r$0 with $r$1. Every unlabeled rank-$r$2 matroid $r$3 on $r$4 admits a unique decomposition

$r$5

where $r$6 is the symmetrized indicator function of the base polytope. The coefficient vector

$r$7

is therefore well defined, and one sets

$r$8

Its vertices are the extremal matroids (Bonin, 19 Jul 2025).

An equivalent description uses Schubert matroids. Let $r$9 be the set of isomorphism classes of Schubert matroids of rank $n$0 on $n$1 elements. For each matroid $n$2, the symmetrized indicator function of its base polytope has a unique Schubert-expansion

$n$3

and one writes

$n$4

Then

$n$5

Möbius inversion in the cyclic-chain lattice gives $n$6 for every $n$7, so the polytope lies in the affine hyperplane $n$8 (Collins et al., 12 May 2026).

Ferroni and Fink also describe a basis-indicator model. If $n$9 and $\Omega_{n,r}$0, each rank-$\Omega_{n,r}$1 matroid $\Omega_{n,r}$2 on $\Omega_{n,r}$3 defines a $\Omega_{n,r}$4-$\Omega_{n,r}$5 vector $\Omega_{n,r}$6 recording which $\Omega_{n,r}$7-subsets are bases. The labeled polytope

$\Omega_{n,r}$8

has dimension $\Omega_{n,r}$9. Passing to orbit-sums under the symmetric-group action produces the unlabeled quotient $\Omega_{r,n}$0. The Schubert-expansion polytope is the refined coordinate model in which valuative invariants become linear functionals (Ferroni et al., 27 Feb 2025).

2. Valuative invariants and the characterization of vertices

The central structural notion is that of a valuative invariant. A real-valued matroid invariant $\Omega_{r,n}$1 is valuative if it factors through the Derksen–Fink indicator-function decomposition. Bonin records several equivalent descriptions: $\Omega_{r,n}$2 may be expressed as a linear combination of the coefficients $\Omega_{r,n}$3 in the universal $\Omega_{r,n}$4-invariant, or of the Tutte-polynomial coefficients $\Omega_{r,n}$5, or of the flat-counting invariants $\Omega_{r,n}$6 (Bonin, 19 Jul 2025).

Ferroni and Fink’s Lemma 4.3 gives the basic vertex criterion. Suppose $\Omega_{r,n}$7 is a finite sequence of real-valued valuative invariants. One considers the isomorphism types of rank-$\Omega_{r,n}$8 matroids on $\Omega_{r,n}$9 and, at each stage, retains only those matroids maximizing $2$0 among the survivors. If the remaining isomorphism types all determine the same point $2$1, then $2$2 is a vertex of $2$3. Bonin emphasizes that, in practice, one alternates maximization and minimization of valuative invariants in order to isolate either a single extremal matroid or a small class collapsing to one vertex (Bonin, 19 Jul 2025).

This principle reinterprets extremal matroid theory polyhedrally. Classical combinatorial optimization statements such as maximizing the number of $2$4-point lines, maximizing the number of large hyperplanes, or maximizing selected flag counts become linear optimization over $2$5. A family is extremal precisely when a short chain of valuative optimizations singles out its Schubert-expansion point. This suggests a systematic bridge between convex geometry and the older program of characterizing matroids by extremal counting data.

3. Infinite families of extremal matroids

Bonin proves that many matroids conjectured by Ferroni and Fink to yield vertices do in fact determine vertices of the unlabeled polytope. The proofs all proceed by exhibiting short sequences of valuative-invariant optimizations that isolate the target family (Bonin, 19 Jul 2025).

Family	Extremality statement	Source
Cycle matroids of complete graphs	$2$6 is extremal	(Bonin, 19 Jul 2025)
Truncations of $2$7	$2$8 is extremal for $2$9	(Bonin, 19 Jul 2025)
Bose–Burton geometries	All $3$0 are extremal	(Bonin, 19 Jul 2025)
Perfect matroid designs	Any perfect matroid design is extremal	(Bonin, 19 Jul 2025)
Dowling geometries	All rank-$3$1 Dowling geometries $3$2 are extremal	(Bonin, 19 Jul 2025)
Spikes with tips	Free spikes, binary spikes, and $3$3-spikes are extremal	(Bonin, 19 Jul 2025)
Direct sums of uniforms	Direct sums of uniform matroids are extremal	(Bonin, 19 Jul 2025)

For cycle matroids of complete graphs, Theorem 7.3 states that if $3$4 is a simple rank-$3$5 matroid on $3$6 points, all lines have size $3$7 or $3$8, all planes have size $3$9 or $0 \le r \le n$0, and every rank-$0 \le r \le n$1 flat for $0 \le r \le n$2 has size at most $0 \le r \le n$3, then $0 \le r \le n$4 has at most $0 \le r \le n$5 three-point lines, with equality if and only if $0 \le r \le n$6. Theorem 7.5 gives a flag-invariant variant: if $0 \le r \le n$7 counts full flags whose rank-$0 \le r \le n$8 flat has $0 \le r \le n$9 points, then $r$0, with equality if and only if $r$1.

For Bose–Burton geometries, Theorem 6.4 identifies a hyperplane-count extremal characterization. For Dowling geometries, Theorems 8.2 and 8.4 show that a rank-$r$2 simple matroid with prescribed flat sizes has at most $r$3 hyperplanes of maximum size, with equality characterizing $r$4 for a group of order $r$5; when $r$6, quasigroups also appear. Bonin also notes that all Dowling geometries of fixed rank $r$7 and fixed group order $r$8 map to the same vertex of $r$9.

The spike results are similarly sharp. For $[n]$0, Theorems 9.1 and 9.2 characterize certain spikes with tips by maximizing the number of large lines or the number of large flags. The direct-sum theorems extend the list from connected to disconnected extremal families: direct sums of uniform matroids yield vertices, and more generally direct sums of connected extremal matroids satisfying a flag-counting hypothesis and some additive invariant constraints also yield vertices.

4. Explicit constructions in ranks 2 and 3

Collins and Schleis give recursive constructions for $[n]$1 for $[n]$2 and $[n]$3 for $[n]$4, together with Schubert expansions for all isomorphism classes of rank-$[n]$5 matroids up to $[n]$6 and rank-$[n]$7 matroids up to $[n]$8 (Collins et al., 12 May 2026).

In rank $[n]$9, there are $r$00 Schubert matroids, so Collins and Schleis state $r$01. The polytope decomposes into three blocks corresponding to loopless-coloopless matroids, matroids with at least one loop, and matroids with at least one coloop. The loop block is obtained by adding a zero row to the matrix for $r$02, the coloop block by appending a zero column, and the central block by an explicit recursion involving matrices $r$03 and $r$04. The columns of the resulting matrix $r$05 are exactly the Schubert-expansion vectors $r$06 for rank-$r$07 matroids on $r$08 elements.

In rank $r$09, there are $r$10 Schubert matroids, so Collins and Schleis state $r$11. Again there is a loopful block, a coloopful block, and a central block. The key structural input is that every loopless, coloopless rank-$r$12 matroid is determined by its set of inseparable cyclic flats of rank $r$13, each isomorphic to some $r$14. If $r$15 has $r$16 such flats of sizes $r$17, then its only nonzero Schubert coefficients are

$r$18

General loopless-coloopless rank-$r$19 matroids are recovered from their simplifications by reintroducing parallel classes, and Collins and Schleis obtain explicit update rules for all $r$20 in four insertion cases. Algorithm 4.1 packages these updates into a Julia/OSCAR computation of the full matrix $r$21.

The low-dimensional cases are completely explicit. For $r$22, all four rank-$r$23 matroids are Schubert, so $r$24 and $r$25 is the simplex $r$26. For $r$27, there are $r$28 rank-$r$29 isomorphism classes and $r$30 Schubert matroids, and the full $r$31 matrix $r$32 is written out in the paper. The number of distinct Schubert-expansions in rank $r$33 grows rapidly:

$r$34

This makes higher-rank computation difficult even when the recursion is conceptually uniform.

5. Faces, representative examples, and valuative applications

The unlabeled polytope is not only a vertex classifier; it also organizes substantial families of matroids as faces. Ferroni and Fink show that loopless matroids form a face cut out by minimizing the number of loops, while simple matroids arise by additionally minimizing the number of parallel pairs. Paving matroids are obtained by maximizing the number of independent $r$35-subsets, and the corresponding face has dimension $r$36 in $r$37. Sparse-paving matroids form the intersection of the paving and copaving faces and yield an edge in $r$38. Elementary-split matroids also form a face, with dimension $r$39 in $r$40. The same general method applies to transversal, modular, series-parallel, graphic, cographic, and positroid classes (Ferroni et al., 27 Feb 2025).

Several small examples clarify the geometry. In rank $r$41 on $r$42 points, there is only one matroid, $r$43, and $r$44 is a single point with coordinate $r$45. In rank $r$46 on $r$47 points, the two nonisomorphic matroids are $r$48 and the matroid with one $r$49-point line; these determine the two vertices of the line segment $r$50. In rank $r$51 on $r$52 points, one may compute the $r$53-vectors in $r$54 and verify that the vertices are exactly $r$55, $r$56, and the unique $r$57-spike on $r$58 points (Bonin, 19 Jul 2025).

A particularly notable application concerns representability. Ferroni and Fink show that there exist explicit linear combinations of Tutte-polynomial coefficients, hence valuative invariants, that are nonnegative on every representable matroid but negative at some vertex of $r$59 occupied by a non-representable sparse-paving matroid. This demonstrates that nonnegativity on representable matroids does not extend automatically to all matroids, and it places the obstruction at an extremal point of the unlabeled polytope (Ferroni et al., 27 Feb 2025).

6. Facets, duality, and open problems

The facet structure of the polytope remains largely unknown. Bonin summarizes the current state by noting that Ferroni and Fink give the general dimension formula

$r$60

but that an explicit description of the facet inequalities is open. There is no complete list of inequivalent facets even for small $r$61, and it is unknown whether $r$62 is neighborly or what its full face poset looks like. Bonin also states that no nontrivial inequalities cutting out $r$63 are known beyond those arising from valuative-invariant maximization (Bonin, 19 Jul 2025).

In low rank, facets can in principle be obtained by dualizing the $r$64-description. Collins and Schleis note that in practice one uses a polyhedral package such as OSCAR; for example, the rank-$r$65 construction produces $r$66 columns for $r$67. They also emphasize that the facets come in $r$68-orbits, reflecting the symmetry inherited from relabeling. For $r$69 in rank $r$70, exact numbers of vertices and facets can be extracted computationally, whereas for $r$71 the hull computation is already heavy (Collins et al., 12 May 2026).

Several structural conjectures frame the present research program. Ferroni and Fink conjecture that any direct sum of vertices is again a vertex, and Bonin’s direct-sum theorems provide supporting evidence for large restricted families. The behavior under duality is also only partly understood: $r$72 is obtained from $r$73 by a known involution, but the geometry of $r$74 versus $r$75 is mostly unexplored. Further questions concern explicit classification of all vertices, asymptotic bounds on the number of extremal classes, and possible connections with Ehrhart theory, motivated by the lattice-point structure of nested-matroid polytopes (Bonin, 19 Jul 2025, Ferroni et al., 27 Feb 2025).

Taken together, these developments place the polytope of all unlabeled matroids at the intersection of matroid theory, polyhedral geometry, and valuation theory. Its current significance lies less in a completed structural theory than in a precise and increasingly productive framework: extremal matroids become vertices, classical counting problems become linear programs, and low-rank computation supplies concrete data against which general conjectures can be tested.