A path model for MV polytopes in type A_n

TL;DR

This paper introduces a new path model for MV polytopes in type A_n, linking Minkowski sums, cluster variables, and folded galleries to provide a combinatorial framework for these polytopes.

Contribution

It develops a one-skeleton path model for MV polytopes, connecting Minkowski sums, cluster variables, and folded galleries in type A_n.

Findings

Minkowski sum of MV polytopes corresponds to path concatenation.

Fundamental paths induce Harder-Narasimhan polytopes.

Paths parameterize cluster variables and relate to folded galleries.

Abstract

We introduce a one-skeleton path model for Mirkovic-Vilonen polytopes in type A_n. We prove that the Minkowski sum of (MV) polytopes corresponds to the concatenation of one-skeleton paths of this model. We show that MV polytopes induced by fundamental one-skeleton paths are Harder-Narasimhan polytopes. The paths given by an orientation of the fundamental alcove parameterize precisely the cluster variables in the initial seed of the coordinate ring C[N]. We also establish a correspondence between fundamental one-skeleton paths and folded galleries representing maximal faces of subword complexes. Under this correspondence, the comultiplication structure of C[N] matches the intrinsic comultiplication structure of folded galleries given by projections to sub-Coxeter complexes.