Characterization of neighboring vertices

Characterize when two matroids yield neighboring vertices of \overline{\Omega}_{r,n}, and when two isomorphism classes of extremal matroids yield neighboring vertices of \Omega_{r,n}.

Background

After analyzing vertices and several faces, the paper identifies the edge structure as a substantially less understood aspect of the two polytopes. The question asks for adjacency criteria both in the labelled polytope and in the unlabelled polytope, where only extremal matroids correspond to vertices.

References

When do two matroids $M$ and $N$ give a pair of neighboring vertices of $\overline{\Omega}{r,n}$? When do two isomorphism classes of extremal matroids yield neighboring vertices of $\Omega{r,n}$?

— The polytope of all matroids  (2502.20157 - Ferroni et al., 27 Feb 2025) in Section 6.3, “Edges of the polytopes”