Characterizations of certain matroids by maximizing valuative invariants

TL;DR

The paper investigates how to identify vertices of the unlabeled matroid polytope $\Omega_{r,n}$ by maximizing sequences of valuative invariants. It proves that a number of matroids conjectured to be extremal (including cycle matroids of complete graphs, projective and Dowling geometries) are indeed extremal, and extends the results to truncations, Bose–Burton geometries, spikes, and direct sums of uniform matroids. A general framework is developed showing direct sums of certain extremal matroids remain extremal under additive valuative invariants, connecting both classical and novel extremal families. Central to the approach is the universal valuative invariant $\mathcal{G}$, which allows translating vertex characterization into optimization problems over valuative invariants, yielding concrete identifications of extremal matroids across several families.

Abstract

Luis Ferroni and Alex Fink recently introduced a polytope of all unlabeled matroids of rank $r$ on $n$ elements, and they showed that the vertices of this polytope come from matroids that can be characterized by maximizing a sequence of valuative invariants. We prove that a number of the matroids that they conjectured to yield vertices indeed do (these include cycle matroids of complete graphs, projective geometries, and Dowling geometries), and we give additional examples (including truncations of cycle matroids of complete graphs, Bose-Burton geometries, and binary and free spikes with tips). We prove a special case of a conjecture of Ferroni and Fink by showing that direct sums of uniform matroids yield vertices of their polytope, and we prove a similar result for direct sums whose components are in certain restricted classes of extremal matroids.

Figures (3)

• Figure 1: A matroid $M$, its lattice $\mathcal{Z}(M)$ of cyclic flats, and the lattice $\mathcal{C}(\mathcal{Z}(M))$ of chains of cyclic flats that include $\emptyset$ and $E(M)$, with $\mu(C,\hat{1})$ shown beneath $C$. For readability, we suppress $\emptyset$ and $E(M)$ in the chains and let $A=\{a,a'\}$, $B=\{b,b'\}$, $C=\{c,c'\}$, and $D=\{d,d'\}$.
• Figure 2: A second example of computing the coefficients $-\mu(C,\hat{1})$ in Equation (\ref{['eq:lcnested']}) following the same conventions as in Figure \ref{['fig:MuExample1']}.
• Figure 3: The Dowling geometry $Q_3(G)$ where $G$ is the sign group, $\{1,-1\}$.

Theorems & Definitions (38)

• Lemma 2.1
• Corollary 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Lemma 2.7
• Theorem 3.1
• proof
• Theorem 4.1