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

TL;DR

This paper decomposes tensor products of certain affine Lie algebra modules and derives generating function identities, revealing new partition identities for specific cases.

Contribution

It provides a detailed decomposition of $ abla$-modules and introduces novel partition identities for $n=3,4$.

Findings

Decomposition of $ abla$-modules tensor products.

New partition identities for $n=3,4$.

Generating function identities for outer multiplicities.

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 $n=3, 4$.