The crystal structure on the set of Mirkovic-Vilonen polytopes

By Joel Kamnitzer. 2005

85 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/572162147546d972f91548b99dfc593315d2efb6

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Preprojective algebras and MV polytopes (2010, background): “On the combinatorial side, a crystal can be realized by several models, each of which provides a concrete access to it; as an example, Anderson [1] and the second author’s [10, 11] MV (Mirković, Vilonen) polytopes form a model for the crystal B(−∞) of the positive part U(n) of U(g).”
Mirković–Vilonen polytopes and Khovanov–Lauda–Rouquier algebras (2012, background): “It is shown in [Kam10, Kam07] that a α (P ) also satisfies (i), from which it is immediate that a α (P ) = a α (b).”