The crystal structure on the set of Mirkovic-Vilonen polytopes

TL;DR

This paper demonstrates the equivalence of two crystal structures on MV polytopes, confirms a conjecture for sl_n, provides a counterexample for sp_6, and links Kashiwara data to MV polytopes.

Contribution

It establishes the agreement of crystal structures on MV polytopes and verifies a conjecture, advancing understanding of their combinatorial and geometric properties.

Findings

Crystal structures on MV polytopes are shown to agree.

Conjecture of Anderson-Mirkovic is proved for sl_n.

Counterexample found for sp_6 crystal structure.

Abstract

In an earlier work, we proved that MV polytopes parameterize both Lusztig's canonical basis and the Mirkovic-Vilonen cycles on the Affine Grassmannian. Each of these sets has a crystal structure (due to Kashiwara-Lusztig on the canonical basis side and due to Braverman-Finkelberg-Gaitsgory on the MV cycles side). We show that these two crystal structures agree. As an application, we consider a conjecture of Anderson-Mirkovic which describes the BFG crystal structure on the level of MV polytopes. We prove their conjecture for sl_n and give a counterexample for sp_6. Finally we explain how Kashiwara data can be recovered from MV polytopes.