Limits of Multiplicities in Excellent Filtrations and Tensor Product Decompositions for Affine Kac-Moody Algebras

TL;DR

This paper investigates the multiplicities in tensor product decompositions of affine Kac-Moody algebra modules, expressing them via Demazure filtrations, and derives new partition identities for specific cases.

Contribution

It provides a novel method to compute tensor product multiplicities using excellent filtrations and applies this to derive explicit formulas and identities for affine algebra modules.

Findings

Expressed multiplicities as sums over Demazure flags.

Derived explicit formulas for tensor products of fundamental modules.

Established new partition identities related to affine algebra representations.

Abstract

We express the multiplicities of the irreducible summands of certain tensor products of irreducible integrable modules for an affine Kac-Moody algebra over a simply laced Lie algebra as sums of multiplicities in appropriate excellent filtrations (Demazure flags). As an application, we obtain expressions for the outer multiplicities of tensor products of two fundamental modules for $\widehat{\mathfrak{sl}}_2$ in terms of partitions with bounded parts, which subsequently lead to certain partition identities.