The Verlinde ring of a (let us say) simply connected simple compact Lie group has as underlying additive group the Grothendieck group of representations of the central extension $\widehat{LG}$ of the loop of $G$ with the 'positive energy' condition. I'm trying to find a concise mathematical definition of the product on this ring, the so-called fusion product. In Freed-Hopkins-Teleman's Loop groups and twisted K-theory III they define an $R(G)$-module structure on this additive group using induction of representations (hence a functorial description).
All references I'm reading just mention 'fusion rules', and cite Verlinde, whose 1988 paper is in the journal Nuclear Physics B. Surely we have a more recent discussion along the lines of the FHT construction mentioned above, and not in terms of linear combinations of coefficients of some irreps considered as generators...?