MV-Polytopes via affine buildings

TL;DR

This paper presents a new geometric and combinatorial construction of MV-polytopes for complex semisimple groups using affine buildings, Bott-Samelson varieties, and LS-galleries, linking algebraic and geometric structures.

Contribution

It introduces an explicit geometric and combinatorial method to construct MV-polytopes via affine buildings and LS-galleries, expanding the understanding of their structure.

Findings

Constructed MV-polytopes using affine building geometry.

Linked MV-cycles to LS-galleries and GGMS strata.

Provided a combinatorial description using crystal structures.

Abstract

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.