Rank 2 affine MV polytopes

Abstract

We give a realization of the crystal B ( − ∞ ) B(-\infty ) for s l ^ 2 \widehat {\mathrm {sl}}_2 using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito’s characterization of B ( − ∞ ) B(-\infty ) , in terms of the ∗ * involution. The polygons we use have combinatorial properties suggesting they are the s l ^ 2 \widehat {\mathrm {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 ) A_2^{(2)} , the other rank 2 affine Kac-Moody algebra.