Fusion Rings for Quantum Groups

We study the fusion rings of tilting modules for a quantum group at a root of unity modulo the tensor ideal of negligible tilting modules. We identify them in type A with the combinatorial rings from Korff, C., Stroppel, C.: The -WZNW fusion ring: a combinato-rial construction and a realisation as quotient of quantum cohomology. Adv. Math. 225(1), 200-268, (2010) and give a similar description of the -fusion ring in terms of non-commutative symmetric functions. Moreover we give a presentation of all fusion rings in classical types as quotients of polynomial rings. Finally we also compute the fusion rings for type G (2).