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Mirković–Vilonen polytopes and Khovanov–Lauda–Rouquier algebras

Part of: Representation theory of rings and algebras Grothendieck groups and $K_0$

Published online by Cambridge University Press:  22 June 2016

Peter Tingley  and
Ben Webster
Peter Tingley
Affiliation:
Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL, USA email ptingley@luc.edu
Ben Webster
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA, USA email bwebster@virginia.edu
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Abstract

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We describe how Mirković–Vilonen (MV) polytopes arise naturally from the categorification of Lie algebras using Khovanov–Lauda–Rouquier (KLR) algebras. This gives an explicit description of the unique crystal isomorphism between simple representations of KLR algebras and MV polytopes. MV polytopes, as defined from the geometry of the affine Grassmannian, only make sense in finite type. Our construction on the other hand gives a map from the infinity crystal to polytopes for 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 the 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 Baumann, Kamnitzer, and the first author in symmetric affine types. We also briefly discuss the situation beyond affine type.

Keywords

Mirković–Vilonen polytopesKLR algebrascrystals

MSC classification

Primary: 16G10: Representations of Artinian rings
Secondary: 19A49: $K_0$ of other rings
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 8 , August 2016 , pp. 1648 - 1696
Copyright
© The Authors 2016 

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