Polytope of matroid classes in Hampe’s intersection ring

Study the convex hull of the classes of loopless matroids in Hampe’s intersection ring, equivalently the polytope formed by the Schubert-expansion vectors of loopless matroids.

Background

Hampe introduced an intersection ring of matroids in which only loopless matroids determine classes. The paper observes that these classes expand into loopless Schubert matroid classes with the same coefficients used to define the labelled polytope of all matroids. It identifies the resulting convex hull as a face of the paper’s polytope, while recording that its systematic study had previously been left open.

References

In Section~6.2 of his paper introducing the intersection ring of matroids, Hampe left the following open problem: study the polytope that arises as the convex hull of all matroid classes in that ring.

— The polytope of all matroids  (2502.20157 - Ferroni et al., 27 Feb 2025) in Remark following Proposition 6.?, Section 6.1, subsection “Simple and disconnected matroids”