Mirkovic-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras

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.

57 pages; v3: another significant revision. Results are basically the same, but proofs are reorganized; v4: minor fixes, typos, etc; v5: corrected small error in formula for twisted affine cases

References in corpus (6)

• 2-Kac-Moody algebras
• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Knot invariants and higher representation theory
• Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras II
• Representations of Khovanov-Lauda-Rouquier algebras III: Symmetric Affine Type
• Crystal bases and two-sided cells of quantum affine algebras

Cited by in corpus (23)

• Canonical bases and higher representation theory
• Affine highest weight categories and affine quasihereditary algebras
• Disordered skyrmion phase stabilized by magnetic frustration in a chiral magnet
• Combinatorial descriptions of the crystal structure on certain PBW bases
• Quivers with relations for symmetrizable Cartan matrices IV: Crystal graphs and semicanonical functions
• Rigged configurations and the $\ast$-involution
• PBW parametrizations and generalized preprojective algebras
• Representations of Khovanov-Lauda-Rouquier algebras III: Symmetric Affine Type
• Cuspidal systems for affine Khovanov-Lauda-Rouquier algebras
• Laurent phenomenon and simple modules of quiver Hecke algebras
• Combinatorial descriptions of the crystal structure on certain PBW bases (extended abstract)
• Folding KLR algebras
• Young tableaux, multi-segments, and PBW bases
• Crystal bases and three-dimensional $\mathcal{N}=4$ Coulomb branches
• Affinizations, R-matrices and reflection functors
• Modular Representation Theory of Symmetric Groups
• Affine PBW Bases and MV Polytopes in Rank 2
• Rigged configurations and the $\ast$-involution for generalized Kac--Moody algebras
• A skew Specht perspective of RoCK blocks and cuspidal systems for KLR algebras in affine type A
• Categorification of the internal braid group action for quantum groups I: 2-functoriality
• Alcove path model for $B(\infty )$
• Affine PBW Bases and Affine MV polytopes
• Morita equivalences between cyclotomic KLR algebras in types ${\mathtt{C}}_{\infty }$ and ${\mathtt{A}}_{\infty }$