MV-Polytopes via affine buildings

Details MV-Polytopes via affine buildings MV-Polytopes via affine buildings

2009-03-23

arxiv.org/abs/0903.3744v1

Mathematics

Representation Theory

34 pages

Scientific paper

We give a construction of MV-polytopes of a complex semisimple algebraic group G in terms of the geometry of the Bott-Samelson variety and the affine building. This is done by using the construction of dense subsets of MV-cycles by Gaussent and Littelmann. They used LS-gallery to define subsets in the Bott-Samelson variety that map to subsets of the affine Grassmannian, whose closure are MV-cycles. Since points in the Bott-Samelson variety correspond to galleries in the affine building one can look at the image of a point in such a special subset under all retractions at infinity. We prove that these images can be used to construct the corresponding MV-polytope in an explicit way, by using the GGMS strata. Furthermore we give a combinatorial construction for these images by using the crystal structure of LS-galleries and the action of the ordinary Weyl group on the coweight lattice.

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