Duke Mathematical Journal

MV-polytopes via affine buildings

Michael Ehrig

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For an algebraic group G, Anderson introduced the notion of Mirković-Vilonen (MV) polytopes as images of MV-cycles under the moment map of the affine Grassmannian. It was shown by Kamnitzer that MV-polytopes and their corresponding cycles can be described as solutions of the tropical Plücker relations. Another construction of MV-cycles, by Gaussent and Littelmann, can be given by using LS-galleries, a more discrete version of Littelmann's path model.

This article gives a direct combinatorial construction of the MV-polytopes using LS-galleries. This construction is linked to the retractions of the affine building and the Bott-Samelson variety corresponding to G, leading to a type-independent definition of MV-polytopes not involving the tropical Plücker relations.

Article information

Duke Math. J., Volume 155, Number 3 (2010), 433-482.

First available in Project Euclid: 16 November 2010

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 17B10: Representations, algebraic theory (weights)


Ehrig, Michael. MV-polytopes via affine buildings. Duke Math. J. 155 (2010), no. 3, 433--482. doi:10.1215/00127094-2010-062. https://projecteuclid.org/euclid.dmj/1289916770

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