Stable properties of plethysm : on two conjectures of Foulkes

Two conjectures made by II.O. Foulkes in 1950 can be stated as follows.

1)

Denote byV a finite-dimensional complex vector space, and byS m V itsm-th symmetric power. Then the GL(V)-moduleS n (S m V ) contains the GL(V)-moduleS n (S m V ) forn > m.

2)

For any (decreasing) partition λ = (λ1,λ2,λ3,...), denote byS λ V the associated simple, polynomial GL(V)-module. Then the multiplicity of in the GL(V)-moduleS n (S m+p Y) is an increasing function ofp. We show that Foulkes' first conjecture holds forn large enough with respect tom (Corollary 1.3). Moreover, we state and prove two broad generalizations of Foulkes' second conjecture. They hold in the framework of representations of connected reductive groups, and they lead e.g. to a general analog of Hermite's reciprocity law (Corollary 1 in 3.3).