Stable properties of plethysm : on two conjectures of Foulkes

abstract

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$S_{(\lambda _1 + np,\lambda _2 \lambda _{3,...} )} V$ 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).