Two generalizations of the PRV conjecture

Abstract

Let G be a complex connected reductive group. The Parthasarathy-Ranga RaoVaradarajan (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.

Generalizations of the PRV conjecture, II

“…In particular, this result is a common generalization of the two results of [MPR11] explained before.…”

“…This paper is a continuation of [MPR11] where the authors obtained a new proof and two generalizations of the PRV conjecture.…”

“…ofĜ). Letν ∈ X(T ) + ; in [MPR11], we proved that for anyŵ ∈Ŵ , V G (ρ(ŵν)) is a submodule of VĜ(ν). Such a submodule of VĜ(ν) is called a PRV component.…”

“…Let µ and ν be two dominant weights of G. In [MPR11], we also improve the PRV conjecture by constructing a family of irreducible submodules of V G (µ) ⊗ V G (ν), strictly containing the PRV components. The cardinality of this family is not bounded independently of the couple (λ, µ).…”

“…In [MPR11], we generalized the PRV conjecture: for any irreduciblê G-moduleV , we give an explicit family of irreducible G-submodules ofV parametrized by W \Ŵ .…”