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Stable properties of plethysm: On two conjectures of Foulkes. (English) Zbl 0823.20039
Summary: Two conjectures made by H. O. Foulkes in 1950 can be stated as follows. 1) Denote by \(V\) a finite-dimensional complex vector space, and by \(S_ m V\) its \(m\)-th symmetric power. Then the \(\text{GL} (V)\)-module \(S_ n (S_ m V)\) contains the \(\text{GL}(V)\)-module \(S_ m (S_ n V)\) for \(n > m\). 2) For any (decreasing) partition \(\lambda = (\lambda_ 1, \lambda_ 2, \lambda_ 3, \dots)\), denote by \(S_ \lambda V\) the associated simple, polynomial \(\text{GL} (V)\)-module. Then the multiplicity of \(S_{(\lambda_ 1 + np, \lambda_ 2, \lambda_ 3, \dots)} V\) in the \(\text{GL}(V)\)-module \(S_ n (S_{m + p} V)\) is an increasing function of \(p\). We show that Foulkes’ first conjecture holds for \(n\) large enough with respect to \(m\) (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).

MSC:
20G05 Representation theory for linear algebraic groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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