Mirkovic-Vilonen cycles and polytopes in type A

Details Mirkovic-Vilonen cycles and polytopes in type A Mirkovic-Vilonen cycles and polytopes in type A

2003-05-21

arxiv.org/abs/math/0305297v3

Mathematics

Algebraic Geometry

updated title and introduction

Scientific paper

We study, in type A, the algebraic cycles (MV-cycles) discovered by I. Mirkovi\'c and K. Vilonen [MV]. In particular, we partition the loop Grassmannian into smooth pieces such that the MV-cycles are their closures. We explicitly describe the points in each piece using the lattice model of the loop Grassmannian in type A. The partition is invariant under the action of the coweights and, up to this action, the pieces are parametrized by the Kostant parameter set. This description of MV-cycles allows us to prove the main result of the paper: the computation of the moment map images of MV-cycles (MV-polytopes) by identifying the vertices of each polytope.

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