fusionCoefficient -- computes the multiplicity of W in the fusion product of U and V

This function implements the Kac-Walton algorithm; see Di Francesco, Mathieu, and Senechal, Conformal Field Theory, Springer Graduate Texts in Theoretical Physics, Section 16.2.2.

Given three irreducible Lie algebra modules $U$, $V$, and $W$, the function returns the multiplicity of $W$ in the fusion product of $U$ and $V$ at level $l$. (We are abusing notation and terminology a little here; the fusion product is really a product for modules over an affine Lie algebra. However, since the Kac-Walton algorithm is defined entirely using the combinatorics of the root system of the underlying finite-dimensional Lie algebra, we may therefore use the Kac-Walton algorithm to define a product on Lie algebra modules as well.)

The example below shows that for $g=sl_3$ and $\lambda=2 \omega_1 + \omega_2$, $\mu= \omega_1 + 2 \omega_2$, and $\nu= \omega_1 + \omega_2$, the level 3 fusion product $V_{\lambda} \otimes_3 V_{\mu}$ contains one copy of $V_{\nu}$.