Stability of Type A Mirkovi\'c-Vilonen Polytopes under Minkowski Sum via Weak Separation

Abstract

Mirkovi\'c--Vilonen (MV) polytopes play a key role in the representation theory of reductive algebraic groups, while the geometric behavior of prime MV polytopes under Minkowski addition remains a subtle open problem. This paper focuses on type A and regards Schubert matroid polytopes as fundamental prime MV building blocks. Using the crystal structure on MV polytopes, we strengthen Sanchez's compatibility condition and establish a necessary and sufficient condition: the positive Minkowski sum of such polytopes is again an MV polytope precisely when the indexing family is weakly separated. Working within discrete convex analysis, we relate discrete concave tropical Pl\"ucker functions to concave extensions on the hypercube and the resulting generalized matroid subdivisions, showing that weak separation is equivalent to the stability of these subdivisions under common refinement. We further clarify the intrinsic connection between our subdivision constructions and the hypersimplex matroid subdivisions developed by Early, providing a natural flag-type generalization of his classical results. We briefly discuss generalized positroids and generalized polypositroids, and identify the MV fan $\mathcal{MV}$ as the secondary fan of hypercube generalized positroid subdivisions. Accordingly, maximal weakly separated sets correspond to maximal cones in $\mathcal{MV}$ and produce the finest such subdivisions. This work unifies MV polytope theory with tropical matroid geometry, advances the understanding of compatibility phenomena in MV combinatorics, and offers new perspectives at the interface of representation theory and combinatorics.