2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73923We give an explicit description of the Mirkovic-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. We prove that a polytope is an MV polytope if and only if it a lattice polytope whose defining hyperplanes are parallel to those of the Weyl polytopes and whose 2-faces are rank 2 MV polytopes. As an application, we give a bijection between Lusztig's canonical basis and the set of MV polytopes.42 pages, v2: new sections added, references improved, mistakes corrected, exposition improvedAlgebraic GeometryRepresentation Theorytext