PRV for the Fusion product, the case $\lambda\gg\mu$

TL;DR

This paper proves the existence of PRV components in the fusion product of certain representations of complex simple Lie algebras when one weight dominates the other significantly, confirming a special case of a broader conjecture.

Contribution

It establishes the presence of PRV components in the fusion product for the case where \\lambda \\gg \\mu, a previously unverified scenario.

Findings

PRV components exist in the fusion product under the \\lambda \\gg \\mu assumption

Verification of PRV component existence in this special case

Supports broader conjectures about fusion products in Lie algebra representations

Abstract

Given a complex simple Lie algebra $\mathfrak{g}$ and a positive integer $\ell$, under the assumption $\lambda\gg\mu$, we show that irreducible representations of $\mathfrak{g}$ of the form $V(\lambda +w\mu)$, $w\in W,$ with level at most $\ell$ appear in the fusion product of $V(\lambda)$ and $V(\mu)$. This verifies the existence of PRV components for the fusion product in this special case.