M. Brion

Dec 1, 1993

Citations

18

Influential Citations

109

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Journal

manuscripta mathematica

Abstract

abstractTwo conjectures made by II.O. Foulkes in 1950 can be stated as follows.1)Denote byV a finite-dimensional complex vector space, and bySmV itsm-th symmetric power. Then the GL(V)-moduleSn(SmV) contains the GL(V)-moduleSn(SmV) forn > m.2)For any (decreasing) partition λ = (λ1,λ2,λ3,...), denote bySλV the associated simple, polynomial GL(V)-module. Then the multiplicity of $$S_{(\lambda _1 + np,\lambda _2 \lambda _{3,...} )} V$$ in the GL(V)-moduleSn(Sm+pY) 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).

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