Resumen de The crystal structure on the set of Mirkovic-Vilonen polytopes
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 and give a counterexample for . Finally we explain how Kashiwara data can be recovered from MV polytopes.
