"Generators" for fusion rings It's a rather obvious idea in the area of fusion rings, but I haven't found a
reference yet. Start with the usual rules for a rank n fusion ring
$X_i\bigotimes{X_j}=\Sigma_k{T_{ij}}^kX_k$
and interpret the objects $X_i$ as n diagonal matrices $D_{ii}$ and $\bigotimes$ as ordinary matrix multiplication. The (more or less unique) solution of this system is even simpler: Interpret the $X_i$ as scalars and $\bigotimes$ as very ordinary :-) multiplication. $D_{ii}$  is then the i-th solution of this system (having exactly n solutions), placed on the diagonal. You can collect the n diagonals in a final n*n matrix $M_{ij}$. (Some freedom of row/column permutation - make sure that the first row consists only of 1, put the other rows in any order.)
You now can combine $M$ and the Verlinde $S$ matrix to a lot of "cool" equations, e.g. (+ is transpose) $A=MS=(MS)^+$ (A is symmetric) or $B=S^{+-}M$ (B is diagonal). Computing $S$ from $M$ is very easy (even if $S$ is NOT symmetric!).
I called the $D_{ii}$ "generators" because this all resembles, especially in graphic form (Dinotracks - like Birdtracks, only different :-) somewhat Lie algebra generators (you also have a "Jacobi relation" etc.).
Do you know a paper where the matrices M have been put to good use? (I have no idea if computing the Verlinde $S$ matrix from given fusion rules is considered as a "hard" problem, and the feeling that classification of fusion rings works the other way round anyway - restrict $S$ and compute all possible $T$ that remain.)
Can you give a proof for the "cool" equations? (I merely observed them.)
Here are some more if you are interested: http://imgur.com/1ZfBVTm
(Dropping arrows - matrix is symmetric and real, hole in dot - it is also diagonal)
Bonus actual example:
Rule: $A\bigotimes{A}=1,A\bigotimes{B}=B,B\bigotimes{B}=1\bigoplus{A}\bigoplus{B}$
$M=\begin{pmatrix}
 1 & 1 & 1\\
 1 & 1 & -1\\
 2 & -1 & 0
\end{pmatrix}$
$S=\begin{pmatrix}
 1/\surd{6} & 1/\surd{6} & \surd{2/3}\\
 1/\surd{3} & 1/\surd{3} & -1/\surd{3}\\
 1/\surd{2} & -1/\surd{2} & 0
\end{pmatrix}$
(Any chemist will observe M is the character table of $C_{3v}$ and the multiplication table is the same but this is only half an accident :-)
 A: This is just an answer to part of the question, namely how to determine $S$ from the fusion rules in the case of "modular data".
See also chapter 5, here
http://www.theorie.physik.uni-goe.de/papers/rehren/89/braid_group_statistics.pdf
for a comparison between the characters of a group and the $S$-matrix of a unitary modular tensor category.
If you know all fusion matrices, you can in principle calculate the Verlinde matrix $S$. You need that the fusion rules are abelian. Then the matrices $N_i=(T^k_{ij})$ are commuting and you find a common set of eigenvectors $e_i$, i.e.
$$
N_j e_i=\lambda_{j,i} e_i.$$
Let $e_0$ be the vector with the greatest eigenvalue and normalize it such that 
$e_0=(1,d_1,\ldots,d_n)$.
They also fulfill (using Frobenius reciprocity and commutativity)
$$
N_iN_j=\sum_k T_{ij}^k N_k.
$$ 
Applying this to $e_0$ you get 
$$
d_id_j=\sum_k T^k_{ij} d_k,
$$
in other words $d_i$ are the Perron-Frobenius dimensions.
Then the Verlinde matrix should be given as:
$$
S= \frac1{\sqrt{\sum_i d_i^2}}\left(e_0,\frac{e_{\sigma(1)}}{d_1},\ldots,\frac{ e_{\sigma(n)}}{d_n}\right),
$$
where $\sigma$ is a permutation of ${1,\ldots,n}$ and $e_i$ are normalized such that the first entry is 1.
