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...?



Concise mathematical definition of the fusion product on the Verlinde ring?
The fusion product arises from the braided monoidal structure on the modular tensor category of [adjectives] loop group representations at level $k$ itself. This is the value $Z(S^1)$ of Chern-Simons theory with gauge group $G$ and level $k$ on the circle, and accordingly its braided monoidal structure comes from thinking about what Chern-Simons assigns to a pair of pants / a sphere punctured at three points / a disk with two disks inside it, namely a "bilinear functor"


$$\boxtimes : Z(S^1) \times Z(S^1) \to Z(S^1)$$


which, in terms of the "basis" of $Z(S^1)$ given by the irreducible representations at level $k$, is completely specified by giving a vector space for each triple $(V\_i, V\_j, V\_k)$ of irreducible representations, namely $\text{Hom}(V\_i \boxtimes V\_j, V\_k)$. The dimensions of these vector spaces give the "fusion rules" in the Verlinde ring, and can be thought of as being obtained by inserting the representations $V\_i, V\_j, V\_k^{\ast}$ at the three punctures of a sphere punctured at three points (or something like that).


There are as many ways to describe $\boxtimes$ as there are ways to describe what Chern-Simons assigns to surfaces with boundary: geometric quantization, quantum groups, conformal nets... 


If you just want to pass directly to the Verlinde ring then dimensionally reduce everything I said on $S^1$, by which I mean consider $Z(S^1 \times S^1)$ and $S^1$ times a pair of pants, etc. 


