  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.

Algebraic Geometry