2005/05/18 by Joel Kamnitzer, Kamnitzer, Joel · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.math/0505398
openalex publication_date 2005/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 sln and give a counterexample for sp6. Finally we explain how Kashiwara data can be recovered from MV polytopes.