Fusion algebras, symmetric polynomials, orbits of N-groups, and rank-level duality

A method of computing fusion coefficients for Lie algebras of type $A_{n-1}$ on level $k$ was recently developed by A. Feingold and M. Weiner \cite{FW} using orbits of $\mathbb{Z}_n^k$ under the permutation action of $S_k$ on $k$-tuples. They got the fusion coefficients only for n = 2 and 3. We will extend this method to all $n \geq 2$ and all $k \geq 1$. First we show a connection between Young diagrams and $S_k$-orbits of $\mathbb{Z}_n ^k$, and using Pieri rules we prove that this method works for certain specific weights that generate the fusion algebra. Then we show that the orbit method does not work in general, but with the help of the Jacobi-Trudi determinant, we give an iterative method to reproduce all type A fusion products.