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 :-)