fusionProduct -- computes the multiplicities of irreducibles in the decomposition of 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 two irreducible Lie algebra modules $U$ and $V$, the function returns 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 $\lambda=2 \omega_1 + \omega_2 = (2,1)$, $\mu= \omega_1 + 2 \omega_2 = (1,2)$, the level 3 fusion product $V_{(2,1)} \otimes_3 V_{(1,2)}$ contains one copy of $V_{(0,0)}$ and one copy of $V_{(1,1)}$.