The crystal structure on the set of Mirković–Vilonen polytopes

Abstract

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 Anderson-Mirković which describes the BFG crystal structure on the level of MV polytopes. We prove their conjecture for sln and give a counterexample for sp 6 . Finally we explain how Kashiwara data can be recovered from MV polytopes.

Mirković-Vilonen cycles and polytopes

“…ᏹ where Ꮾ denotes the canonical basis, ᏼ denotes the set of MV polytopes, and ᏹ denotes the set of MV cycles. In [Kam07], we show that these bijections are isomorphisms of crystals with respect to the Kashiwara-Lusztig crystal structure on the canonical basis and the Braverman-Finkelberg-Gaitsgory crystal structure on the set of MV cycles. Another important application of our main result is to answer Question 1.…”

“…More generally, this shows that any face of an MV polytope is an MV polytope. It is possible to understand this fact by using the restriction map q J introduced by Braverman-Gaitsgory [BG01] and further discussed [Kam07].…”

On Mirković-Vilonen cycles and crystal combinatorics

“…Combining Theorem 4.7 in [14] with the proof of Theorem 3.1 in [13], one can see that Ξ(t 0 ⊗ b i (n • )) is the closure of A i (n • ). On the other hand, Theorem 4.5 in [13] says that…”

“…In the course of his work on MV polytopes [13,14], Kamnitzer was led to a description of MV cycles similar to the equality Ξ(t 0 ⊗ b) =Ỹ i,c , but starting from the Lusztig parameter of b instead of the string parameter. In Section 4.5, we show that the equality and Kamnitzer's description are in fact equivalent results.…”

A definition of the crystal commutor using Kashiwara’s involution

“…Remark 2.3 All of this combinatorial structure can be seen easily using the MV polytope model [2]. The inclusions ι correspond to translating polytopes.…”