The crystal structure on the set of Mirkovic-Vilonen polytopes
Quantum Algebra
2007-05-23 v2 Algebraic Geometry
Combinatorics
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.
Cite
@article{arxiv.math/0505398,
title = {The crystal structure on the set of Mirkovic-Vilonen polytopes},
author = {Joel Kamnitzer},
journal= {arXiv preprint arXiv:math/0505398},
year = {2007}
}
Comments
27 pages, v2: section on Kashiwara data added