The -WZNW fusion ring: A combinatorial construction and a realisation as quotient of quantum cohomology

A simple, combinatorial construction of the -WZNW fusion ring, also known as Verlinde algebra, is given. As a byproduct of the construction one obtains an isomorphism between the fusion ring and a particular quotient of the small quantum cohomology ring of the Grassmannian Grk,k+n. We explain how our approach naturally fits into known combinatorial descriptions of the quantum cohomology ring, by establishing what one could call a ‘Boson–Fermion-correspondence’ between the two rings. We also present new recursion formulae for the structure constants of both rings, the fusion coefficients and the Gromov–Witten invariants.