---
title: Polytope of Unlabeled Matroids
url: https://www.emergentmind.com/topics/polytope-of-all-unlabeled-matroids
type: topic
---

# Polytope of Unlabeled Matroids

Searching arXiv for the cited papers to ground the article in current literature.
{"query":"arXiv 2507.14720 Characterizations of certain matroids by maximizing valuative invariants"}
{"query":"arXiv 2605.12336 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 $3$ [2502.20157, 2507.14720, 2605.12336].

## 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 $N_1,\dots,N_N$ with $N=\binom{n}{r}$. Every unlabeled rank-$r$ matroid $M$ on $[n]$ admits a unique decomposition
$$
I(\mathcal P(M))=\sum_{i=1}^N a_i(M)\cdot I(\mathcal P(N_i)),
$$
where $I(-)$ is the symmetrized indicator function of the base polytope. The coefficient vector
$$
\chi_M=(a_1(M),a_2(M),\dots,a_N(M))\in \mathbb R^N
$$
is therefore well defined, and one sets
$$
\Omega_{n,r}:=\operatorname{conv}\{\chi_M:M\text{ unlabeled rank-}r\text{ matroid on }[n]\}\subseteq \mathbb R^N.
$$
Its vertices are the extremal matroids [2507.14720].

An equivalent description uses Schubert matroids. Let $\mathcal S_{r,n}$ be the set of isomorphism classes of Schubert matroids of rank $r$ on $n$ elements. For each matroid $M\in\mathcal M_{r,n}$, the symmetrized indicator function of its base polytope has a unique Schubert-expansion
$$
1_{P(M)}(x)=\sum_{S\in\mathcal S_{r,n}} \lambda_S(M)\,1_{P(S)}(x),
$$
and one writes
$$
o_M=(\lambda_S(M))_{S\in\mathcal S_{r,n}}\in\mathbb R^{|\mathcal S_{r,n}|}.
$$
Then
$$
\Omega_{r,n}:=\operatorname{conv}\{o_M:M\in\mathcal M_{r,n}\}\subset \mathbb R^{|\mathcal S_{r,n}|}.
$$
Möbius inversion in the cyclic-chain lattice gives $\sum_{S\in\mathcal S_{r,n}}\lambda_S(M)=1$ for every $M$, so the polytope lies in the affine hyperplane $x_1+\cdots+x_N=1$ [2605.12336].

Ferroni and Fink also describe a basis-indicator model. If $B_r(n,r)=\{B\subseteq [n]:|B|=r\}$ and $N=\binom{n}{r}$, each rank-$r$ matroid $M$ on $[n]$ defines a $0$-$1$ vector $x(M)\in\{0,1\}^N$ recording which $r$-subsets are bases. The labeled polytope
$$
P_{n,r}:=\operatorname{conv}\{x(M):M\text{ a rank-}r\text{ matroid on }[n]\}\subset \mathbb R^N
$$
has dimension $\binom{n}{r}-n$. Passing to orbit-sums under the symmetric-group action produces the unlabeled quotient $Q_{n,r}$. The Schubert-expansion polytope is the refined coordinate model in which valuative invariants become linear functionals [2502.20157].

## 2. Valuative invariants and the characterization of vertices

The central structural notion is that of a valuative invariant. A real-valued matroid invariant $f$ is valuative if it factors through the Derksen–Fink indicator-function decomposition. Bonin records several equivalent descriptions: $f$ may be expressed as a linear combination of the coefficients $g_{\underline r}(M)$ in the universal $G$-invariant, or of the Tutte-polynomial coefficients $t_{i,j}(M)$, or of the flat-counting invariants $F_{h,k}(M;s_h,\dots,s_k)$ [2507.14720].

Ferroni and Fink’s Lemma 4.3 gives the basic vertex criterion. Suppose $f_1,f_2,\dots,f_t$ is a finite sequence of real-valued valuative invariants. One considers the isomorphism types of rank-$r$ matroids on $[n]$ and, at each stage, retains only those matroids maximizing $f_i$ among the survivors. If the remaining isomorphism types all determine the same point $\chi^\ast\in \Omega_{n,r}$, then $\chi^\ast$ is a vertex of $\Omega_{n,r}$. 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 [2507.14720].

This principle reinterprets extremal matroid theory polyhedrally. Classical combinatorial optimization statements such as maximizing the number of $3$-point lines, maximizing the number of large hyperplanes, or maximizing selected flag counts become linear optimization over $\Omega_{n,r}$. 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 [2507.14720].

| Family | Extremality statement | Source |
|---|---|---|
| Cycle matroids of complete graphs | $M(K_{r+1})$ is extremal | [2507.14720] |
| Truncations of $M(K_{r+1})$ | $\operatorname{Tr}^k(M(K_{r+1}))$ is extremal for $r-k\ge 5$ | [2507.14720] |
| Bose–Burton geometries | All $PG(r-1,q)\setminus PG(r-k,q)$ are extremal | [2507.14720] |
| Perfect matroid designs | Any perfect matroid design is extremal | [2507.14720] |
| Dowling geometries | All rank-$r$ Dowling geometries $Q_r(G)$ are extremal | [2507.14720] |
| Spikes with tips | Free spikes, binary spikes, and $(r,r-1,t+1)$-spikes are extremal | [2507.14720] |
| Direct sums of uniforms | Direct sums of uniform matroids are extremal | [2507.14720] |

For cycle matroids of complete graphs, Theorem 7.3 states that if $M$ is a simple rank-$r$ matroid on $\binom{r+1}{2}$ points, all lines have size $2$ or $3$, all planes have size $3,4,$ or $6$, and every rank-$h$ flat for $4\le h\le r$ has size at most $\binom{h+1}{2}$, then $M$ has at most $\binom{r+1}{3}$ three-point lines, with equality if and only if $M\cong M(K_{r+1})$. Theorem 7.5 gives a flag-invariant variant: if $L(M)$ counts full flags whose rank-$i$ flat has $\binom{i+1}{2}$ points, then $L(M)\le (r+1)!/2$, with equality if and only if $M\cong M(K_{r+1})$.

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-$r$ simple matroid with prescribed flat sizes has at most $r$ hyperplanes of maximum size, with equality characterizing $Q_r(G)$ for a group of order $g$; when $r=3$, quasigroups also appear. Bonin also notes that all Dowling geometries of fixed rank $r$ and fixed group order $g$ map to the same vertex of $\Omega_{n,r}$.

The spike results are similarly sharp. For $r\ge 4$, 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 $\Omega_{2,n}$ for $n\le 33$ and $\Omega_{3,n}$ for $n\le 10$, together with Schubert expansions for all isomorphism classes of rank-$2$ matroids up to $n=80$ and rank-$3$ matroids up to $n=11$ [2605.12336].

In rank $2$, there are $|\mathcal S_{2,n}|=\binom{n}{2}$ Schubert matroids, so Collins and Schleis state $\dim \Omega_{2,n}=\binom{n}{2}-1$. 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 $n-1$, the coloop block by appending a zero column, and the central block by an explicit recursion involving matrices $U(n)$ and $V(n)$. The columns of the resulting matrix $O_{2,n}$ are exactly the Schubert-expansion vectors $o_M$ for rank-$2$ matroids on $n$ elements.

In rank $3$, there are $|\mathcal S_{3,n}|=\binom{n}{3}$ Schubert matroids, so Collins and Schleis state $\dim \Omega_{3,n}=\binom{n}{3}-1$. Again there is a loopful block, a coloopful block, and a central block. The key structural input is that every loopless, coloopless rank-$3$ matroid is determined by its set of inseparable cyclic flats of rank $2$, each isomorphic to some $U_{2,k}$. If $M$ has $t$ such flats of sizes $k_1,\dots,k_t$, then its only nonzero Schubert coefficients are
$$
\lambda_{S(1,2,3)}=1-t,\qquad \lambda_{S(1,2,k_i+1)}=1 \text{ for each } i.
$$
General loopless-coloopless rank-$3$ matroids are recovered from their simplifications by reintroducing parallel classes, and Collins and Schleis obtain explicit update rules for all $\lambda_S$ in four insertion cases. Algorithm 4.1 packages these updates into a Julia/OSCAR computation of the full matrix $O_{3,n}$.

The low-dimensional cases are completely explicit. For $n=4$, all four rank-$3$ matroids are Schubert, so $O_{3,4}=I_4$ and $\Omega_{3,4}$ is the simplex $\operatorname{conv}\{e_1,\dots,e_4\}$. For $n=5$, there are $13$ rank-$3$ isomorphism classes and $10$ Schubert matroids, and the full $10\times 13$ matrix $O_{3,5}$ is written out in the paper. The number of distinct Schubert-expansions in rank $3$ grows rapidly:
$$
n=5\to 13,\quad n=6\to 39,\quad n=7\to 109,\quad n=8\to 310,\quad n=9\to 960,\quad n=10\to 3291.
$$
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 $(r-1)$-subsets, and the corresponding face has dimension $n-r$ in $Q_{n,r}$. Sparse-paving matroids form the intersection of the paving and copaving faces and yield an edge in $Q_{n,r}$. Elementary-split matroids also form a face, with dimension $r(n-r)$ in $Q_{n,r}$. The same general method applies to transversal, modular, series-parallel, graphic, cographic, and positroid classes [2502.20157].

Several small examples clarify the geometry. In rank $2$ on $3$ points, there is only one matroid, $U_{2,3}$, and $\Omega_{3,2}$ is a single point with coordinate $(1,-2,1)$. In rank $2$ on $4$ points, the two nonisomorphic matroids are $U_{2,4}$ and the matroid with one $3$-point line; these determine the two vertices of the line segment $\Omega_{4,2}$. In rank $3$ on $5$ points, one may compute the $\chi$-vectors in $\mathbb R^{10}$ and verify that the vertices are exactly $U_{3,5}$, $AG(2,2)$, and the unique $3$-spike on $5$ points [2507.14720].

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 $Q_{3,19}$ 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 [2502.20157].

## 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
$$
\dim \Omega_{n,r}=\binom{n}{r}-1-(\text{number of linear-dependence relations among the }I(\mathcal P(N_i))),
$$
but that an explicit description of the facet inequalities is open. There is no complete list of inequivalent facets even for small $(n,r)$, and it is unknown whether $\Omega_{n,r}$ is neighborly or what its full face poset looks like. Bonin also states that no nontrivial inequalities cutting out $\Omega_{n,r}$ are known beyond those arising from valuative-invariant maximization [2507.14720].

In low rank, facets can in principle be obtained by dualizing the $V$-description. Collins and Schleis note that in practice one uses a polyhedral package such as OSCAR; for example, the rank-$2$ construction produces $53{,}929$ columns for $O_{2,33}$. They also emphasize that the facets come in $S_n$-orbits, reflecting the symmetry inherited from relabeling. For $n\le 8$ in rank $3$, exact numbers of vertices and facets can be extracted computationally, whereas for $n=9,10$ the hull computation is already heavy [2605.12336].

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: $\chi_{M^\ast}$ is obtained from $\chi_M$ by a known involution, but the geometry of $\Omega_{n,r}$ versus $\Omega_{n,n-r}$ 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 [2507.14720, 2502.20157].

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.

Source: https://www.emergentmind.com/topics/polytope-of-all-unlabeled-matroids