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

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 $w$.