# A study of Kostant-Kumar modules via Littelmann paths

## TL;DR

This paper uses Littelmann's path theory to model Kostant-Kumar modules, revealing their structure as concatenations of Lakshmibai-Seshadri paths and providing new decomposition rules for special linear Lie algebras.

## Contribution

It identifies a path model for KK modules within the tensor product path model and introduces a new procedure for permutation determination in Young tableaux.

## Key findings

- Path model for KK modules as concatenations of LS paths
- Decomposition rule for KK modules via Littlewood-Richardson tableaux
- New method for initial element permutation in Young tableaux

## Abstract

We study, by means of Littelmann's theory of paths, Kostant-Kumar modules (KK modules for short), which by definition are certain submodules of the tensor product of two irreducible integrable highest weight representations of a symmetrizable Kac-Moody algebra. Our main result is an identification of a path model for any KK module as a subset of the well known path model for the tensor product consisting of concatenations of Lakshmibai-Seshadri paths. The technical results about extremal elements in Coxeter groups that we formulate and prove en route and the technique of their proofs should be of independent interest. We also discuss the existence of PRV components and generalised PRV components in KK modules. Specialising to the case of the special linear Lie algebra, we record a decomposition rule for KK modules in terms of Littlewood-Richardson tableaux. In this connection, we present a new procedure to determine the permutation that is the initial element of the minimal standard lift of a semi-standard Young tableau. The appendix, necessitated by the derivation of the tableau decomposition rule, deals with standard concatenations of Lakshmibai-Seshadri paths of arbitrary shapes, of which semi-standard Young tableaux form a very special case.