Fusion Algebras via Symmetric Polynomials and orbits of N-Groups

A method of computing fusion coefficients for Lie algebras of type An-1 on level k was recently developed by A. Feingold and M. Weiner [FW] using orbits of Zk n under the permutation action of Sk on k-tuples. They got the fusion coefficients only for n= 2 and 3. We will extend this method to all n≥2 and all k≥1. First we show a connection between Young diagrams and Sk-orbits of Zkn, 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.