Mirkovic-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras

TL;DR

This paper connects Mirkovic-Vilonen polytopes with Khovanov-Lauda-Rouquier algebras, providing a new combinatorial framework that extends MV polytopes to all symmetrizable Kac-Moody algebras and relates to crystal theory.

Contribution

It introduces a construction of KLR polytopes as general-type analogues of MV polytopes, extending their applicability beyond finite-dimensional Lie algebras.

Findings

Provides an explicit crystal isomorphism between simple KLR modules and MV polytopes.

Gives a combinatorial description of decorated polytopes in affine types.

Recovers known affine MV polytopes in symmetric affine cases.

Abstract

We describe how Mirkovic-Vilonen polytopes arise naturally from the categorification of Lie algebras using Khovanov-Lauda-Rouquier algebras. This gives an explicit description of the unique crystal isomorphism between simple representations of the KLR algebra and MV polytopes. MV polytopes, as defined from the geometry of the affine Grassmannian, only make sense for finite dimensional semi-simple Lie algebras, but our construction actually gives a map from the infinity crystal to polytopes in all symmetrizable Kac-Moody algebras. However, to make the map injective and have well-defined crystal operators on the image, we must in general decorate our polytopes with some extra information. We suggest that the resulting KLR polytopes are the general-type analogues of MV polytopes. We give a combinatorial description of the resulting decorated polytopes in all affine cases, and show that this recovers the affine MV polytopes recently defined by Kamnitzer and Baumann and the first author in symmetric affine types. We also briefly discuss the situation beyond affine type.