The Crystal Structure on the Set of Mirković-vilonen Polytopes
نویسنده
چکیده
In an earlier work, we proved that MV polytopes parameterize both Lusztig’s canonical basis and the Mirković-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 AndersonMirković which describes the BFG crystal structure on the level of MV polytopes. We prove their conjecture for SLn and give a counterexample for Sp6.
منابع مشابه
On Mirković-vilonen Cycles and Crystal Combinatorics
Let G be a complex connected reductive group and let G∨ be its Langlands dual. Let us choose a triangular decomposition n−,∨⊕h∨⊕n+,∨ of the Lie algebra of G∨. Braverman, Finkelberg and Gaitsgory show that the set of all Mirković-Vilonen cycles in the affine Grassmannian G ( C((t)) ) /G ( C[[t]] ) is a crystal isomorphic to the crystal of the canonical basis of U(n+,∨). Starting from the string ...
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