  We determine the graded decompositions of fusion products of
finite-dimensional irreducible representations for simple Lie algebras of rank
two. Moreover, we give generators and relations for these representations and
obtain as a consequence that the Schur positivity conjecture holds in this
case. The graded Littlewood-Richardson coefficients in the decomposition are
parametrized by lattice points in convex polytopes and an explicit hyperplane
description is given in the various types.
