Existence of a new family of irreducible components in the tensor product and its applications

TL;DR

This paper proves the existence of a new family of irreducible components in tensor products of modules over Kac-Moody algebras using crystal theory, advancing understanding of Schur positivity and related conjectures.

Contribution

It introduces a new family of irreducible components in tensor products and proves the Schur positivity conjecture for simple Lie algebras under specific weight conditions.

Findings

Existence of new irreducible components in tensor products.

Proof of Schur positivity conjecture in certain cases.

Application to Kac-Moody algebra representations.

Abstract

In this paper, using crystal theory we prove the existence of a new family of irreducible components appearing in the tensor product of two irreducible integrable highest weight modules over symmetrizable Kac-Moody algebras motivated by the Schur positivity conjecture, Kostant conjecture and Wahl conjecture. We also prove Schur positivity conjecture in full generality when the Lie algebra is a simple Lie algebra under the assumption that $\lambda > > \mu$, i.e. if $\lambda$ and $\mu$ are the two dominant weights appearing in the tensor product then $\lambda+w\mu$ is a dominant weight for all the Weyl group elements $w$.