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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:
 [1] Black, S.C., List, R.J.: A note on plethysm. European Journal of Combinatorics 10, 111–112 (1989) · Zbl 0668.20017 [2] Bourbaki, N.: Groupes et alg$$\sim$$bres de Lie, chap. 7 et 8. Hermann, Paris 1975 · Zbl 0329.17002 [3] Carte, C.: Plethysm of elementary functions. Prépublication du LITP 90.34, Paris 1990 [4] Carre,C., Thibon, J.Y.: Plethysm and vertex operators. Adv. in Appl. Math.13, 390–403 (1992) · Zbl 0782.05089 [5] Demazure, M.: A very simple proof of Bott’s theorem. Invent. math.33, 271–272 (1976) · Zbl 0383.14017 [6] Foulkes, H.O.: Concomitants of the quintic and the sextic up to degree four in the coefficients of the ground form. Journal of the London Math. Soc.25, 205–209 (1950) · Zbl 0037.14902 [7] Howe, R.: (GLn, GLm)-duality and symmetric plethysm. Proceedings of the Indian Academy of Sciences (Math. Sciences)97, 85–109 (1987) · Zbl 0705.20040 [8] James, G., Kerber, A.: The representation theory of the symmetric group. Encyclopedia of Mathematics and its applications, vol. 16, Addison Wesley, Reading 1981 · Zbl 0491.20010 [9] Kraft, H.: Geometrische Methoden in der Invariantentheorie. Vieweg, 1985 · Zbl 0669.14003 [10] Kumar, S.: Proof of Wahl’s conjecture on the surjectivity of the Gaussian map for flag varieties, in: Proceedings of the Hyderabad conference on algebraic groups, 275-278, Manoj Prakashan 1991 · Zbl 0795.14027 [11] Luna, D.: Slices étales. Bull. Soc. math. France Mémoire33, 81–105 (1973) [12] Macdonald, I.G.: Symmetric functions and Hall polynomials. Clarendon Press, Oxford 1979 · Zbl 0487.20007 [13] Manivel, L.: Applications de Gauss et pl$$\sim$$thysme. Pr$$\sim$$publication de l’institut Fourier 214, Grenoble 1992 [14] Wahl, J.: Gaussian maps and tensor products of irreducible representations. Manuscripta Math.73, 229–259 (1991) · Zbl 0764.20022 [15] Weintraub, S.H.: Some observations on plethysms. J. of Algebra129, 103–114 (1990) · Zbl 0695.20013 [16] Weyl, H.: The classical groups. Princeton University Press 1946 · Zbl 1024.20502
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