Two generalizations of the PRV conjecture

TL;DR

This paper simplifies and extends the PRV conjecture, providing new insights into the structure of tensor products of irreducible modules for complex reductive groups, including generalizations and additional irreducible components.

Contribution

It simplifies the proof of the PRV conjecture, generalizes it to other branching problems, and identifies new irreducible components arising from similar reasons.

Findings

Simplified proof of the PRV conjecture

Extended the conjecture to broader branching problems

Discovered additional irreducible components in tensor products

Abstract

Let G be a complex connected reductive group. The PRV conjecture, which was proved independently by S. Kumar and O. Mathieu in 1989, gives explicit irreducible submodules of the tensor product of two irreducible G-modules. This paper has three aims. First, we simplify the proof of the PRV conjecture, then we generalize it to other branching problems. Finally, we find other irreducible components of the tensor product of two irreducible G-modules that appear for "the same reason" as the PRV ones.