Rank 2 affine MV polytopes

TL;DR

This paper constructs a combinatorial realization of the infinity crystal for affine sl(2) using decorated polygons, extending the approach to A_2^{(2)} and connecting to MV polytopes in finite type.

Contribution

It provides a new combinatorial model for the infinity crystal of affine sl(2) and A_2^{(2)} using decorated polygons, linking to MV polytopes.

Findings

Realization of the infinity crystal for affine sl(2) via decorated polygons.

Extension of the model to A_2^{(2)} using Kashiwara's similarity.

Polygons exhibit properties analogous to MV polytopes in finite type.

Abstract

We give a realization of the infinity crystal for affine sl(2) using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito's characterization of the infinity crystal in terms of the * involution. The polygons we use have combinatorial properties suggesting they are the analogues in this case of the Mirkovic-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 only other rank two affine Kac-Moody algebra.