Tensor Product Decomposition of $\hat{\mathfrak{sl}}(n)$ Modules and Identities

Abstract

We decompose the $\hat{\mathfrak{sl}}(n)$-module $V(\Lambda_0) \otimes V(\Lambda_i)$ and give generating function identities for the outer multiplicities. In the process we discover some seemingly new partition identities for $n=3, 4$.