Abstract
For an algebraic group , 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 , leading to a type-independent definition of MV-polytopes not involving the tropical Plücker relations.
Citation
Michael Ehrig. "MV-polytopes via affine buildings." Duke Math. J. 155 (3) 433 - 482, 1 December 2010. https://doi.org/10.1215/00127094-2010-062
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