Rank 2 affine MV polytopes

Abstract

Abstract. We give a realization of the crystal B(−∞) for sl 2 using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito's characterization of B(−∞), in terms of the * involution. The polygons we use have combinatorial properties suggesting they are the sl 2 analogues of the Mirković-Vilonen polytopes defined by Anderson and the third author in finite type. Using Kashiwara's similarity of crystals we also give MV polytopes for A (2) 2 , the other rank 2 affine Kac-Moody algebra.

Mirković–Vilonen polytopes and Khovanov–Lauda–Rouquier algebras

“…This map sends b to the corresponding affine MV polytope in the realization of B(−∞) from [BDKT13]. In particular, the image is exactly MV.…”

“…Rank 2 affine MV polytopes. We briefly review the MV polytopes associated to the affine root systems sl 2 and A (2) 2 in [BDKT13], and recall a characterization of the resulting polytopes developed in [MT14].…”

“…In [BDKT13], the first author and collaborators combinatorially define a set MV of decorated pseudo-Weyl polytopes, which they call rank 2 affine MV polytopes. We will not need the details of this construction, but will instead use the following result from [MT14].…”

“…it satisfies the tropical Plücker relations from [Kam10]). • If ∆ F is of affine type, thenF is an MV polytope for that rank 2 affine algebra (either sl 2 or A (2) 2 ) as defined in [BDKT13].…”

“…The description of rank 2 affine MV polytopes in [BDKT13] is combinatorial, so Theorem B gives a combinatorial characterization of KLR polytopes in all affine cases.…”