Mirkovi\'c-Vilonen Polytopes from Combinatorics

Abstract

Mirkovi\'c-Vilonen (MV) polytopes are a class of generalized permutahedra originating from geometric representation theory. In this paper we study MV polytopes coming from matroid polytopes, flag matroid polytopes, Bruhat interval polytopes, and Schubitopes. We give classifications and combinatorial conditions for when these polytopes are MV polytopes. We also describe how the crystal structure on MV polytopes manifests combinatorially in these situations. As a special case, we show that the Newton polytopes of Schubert polynomials and key polynomials are MV polytopes.