Abstract
We give explicit systems of generators of the algebras of invariant polynomials in arbitrary many vector variables for the classical reflection groups (including the dihedral groups). As an application of the results we prove a generalization of Chevalley's restriction theorem for the classical Lie algebras. In the interesting case when the group is of Coxeter typeD n (n≥4) we use higher polarization operators introduced by Wallach. The least upper bound for the degrees of elements in a system of generators turns out to be independent of the number of vector variables. We conjecture that this is also true for the exceptional reflection groups and then sketch a proof for the group of typeF 4.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[Che55] C. Chevalley,Invariants of finite groups generated by reflections, Amer. J. Math.77 (1955), 778–782.
[CH96] M. Clegg, M. Hunziker,Diagonal invariants for the Weyl group of type F 4, in preparation.
[Hum90] J. E. Humphreys,Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics29, Cambridge University Press, 1990.
[Jer89] H. Jerjen,Weitere Ergebnisse zur allgemeinen Vermutung in der Invariantentheorie von endlichen Spiegelungsgruppen, Diplomarbeit, Universität Basel, Basel, Switzerland, 1989.
[Jos96] A. Joseph,On a Harish-Chandra homomorphism, to appear in C. R. Acad. Sc. Paris.
[Kea96] A. Keaton,Commuting varieties and diagonal Weyl group invariants, preprint.
[KN84] A. A. Kirillov, Yu. A. Neretin,The variety A n of n-dimensional Lie algebra structures In: Some Questions in Modern Analysis, Mekh.-Mat. Fak. Moskov. Gos. Univ., Moscow, 1984, pp. 42–56 (in Russian). English translation:-A. A. Kirillov, Yu. A. Neretin, -The variety A n of n-dimensional Lie algebra structures, A.M.S. Transl., (Ser. 2)137 1987, 21–30.
[Kra84] H. Kraft,Geometrische Methoden in der Invariantentheorie, Aspekte der MathematikD1, Vieweg Verlag, Braunschweig, 1984. Russian translation: Х. Крафт,Геомемрические мемоды в меории инварианмов, Москва, Мир, 1987.
[Lun75] D. Luna,Adhérences d'orbite et invariants, Inventiones Math.29 (1995), 231–238.
[Meh88] M. L. Mehta,Basic sets of invariant polynomials for finite reflection groups, Comm. Algebra16 (1988), 1083–1098.
[Ric88] R. Richardson,Conjugacy classes of n-tupels in Lie algebras and algebraic groups, Duke. Math. J.57 (1988), 1–35.
[Ric79],Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio Math.38 (1979), 311–327.
[Wal93] N. R. Wallach,Invariant differential operators on a reductive Lie algebra and Weyl group representations, J. Amer. Math. Soc.6 (1993), 779–816.
[Wey46] H. Weyl,The Classical Groups, Princeton Math. Series1, Princeton University Press, Princeton NJ, 1946. Russian translation: Г. Веёль,Классические группы, их инварианмы и предсмавления, Москва, ИЛ, 1947.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hunziker, M. Classical invariant theory for finite reflection groups. Transformation Groups 2, 147–163 (1997). https://doi.org/10.1007/BF01235938
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01235938