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

TL;DR

This paper decomposes tensor products of certain affine Lie algebra modules and uncovers new partition identities, providing generating functions for multiplicities in the decomposition.

Contribution

It offers a detailed decomposition of $ abla$ modules and introduces novel partition identities for specific cases, advancing understanding of affine Lie algebra representations.

Findings

Decomposition of $V( abla_0) ensor V( abla_0)$ for $ abla = ext{affine } ext{sl}(n)$.

New partition identities for $n=2,3$ cases.

Explicit generating functions for outer multiplicities.

Abstract

We decompose the $\hat{\mathfrak{sl}}(n)$-module $V(\Lambda_0) \otimes V(\Lambda_0)$ and give generating function identities for the outer multiplicities. In the process we discover some seemingly new partition identities in the cases $n=2,3$.