Graded decompositions of fusion products in rank 2

Abstract

We determine the graded decompositions of fusion products of finite-dimensional irreducible representations for simple Lie algebras of rank 2. 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 parameterized by lattice points in convex polytopes, and an explicit hyperplane description is given in the various types.