Mv-polytopes via Affine Buildings

For an algebraic group G, Anderson originally defined the notion of MV-polytopes in [And03], images of MV-cycles, defined in [MV07], under the moment map of the corresponding affine Grassmannian. It was shown by Kamnitzer in [Kam07] and [Kam05] that these polytopes can be described via tropical relations and give rise to a crystal structure on the set of MV-cycles. Another crystal structure can be introduced using LS-galleries which were defined by Gaussent and Littelmann in [GL05], a more discrete version of Littelmann’s path model. The main result of this paper is a direct combinatorial construction of the MV-polytopes using LS-galleries. In addition we link this construction to the retractions of the affine building and the Bott-Samelson variety corresponding to G. This leads to a definition of MV-polytopes not involving the tropical Plücker relations. Herefore it provides a description of the polytopes independent of the type of the algebraic group via the gallery model and affine buildings.