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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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References:
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