Crystals for Kostant-Kumar modules of $\widehat{\mathfrak{sl}_2}$

TL;DR

This paper constructs crystal bases for Kostant-Kumar submodules of tensor products of level 1 representations of the affine Lie algebra fsl_2, using charged partitions, and describes their irreducible decomposition.

Contribution

It introduces a new combinatorial model for crystals of Kostant-Kumar modules in the affine fsl_2 setting, expanding understanding of their structure.

Findings

Crystals for Kostant-Kumar modules are explicitly constructed.

Charged partitions model effectively describes these crystals.

Decomposition into irreducible components is characterized.

Abstract

We consider the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ and the Kostant-Kumar submodules of tensor products of its level 1 highest weight integrable representations. We construct crystals for these submodules in terms of the charged partitions model and describe their decomposition into irreducibles.