Skip to main content
SpringerLink
Log in

When is a ring of torus invariants a polynomial ring?

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Let ρ:T→GL(V) be a finite dimensional rational representation of a torus over an algebraically closed fieldk. We give necessary and sufficient conditions on the arrangement of the weights ofV within the character lattice ofT for the ring of invariants,k[V]T, to have a homogeneous system of parameters consisting of monomials (Theorem 4.1). Using this we give two simple constructive criteria each of which gives necessary and sufficient conditions fork[V]T to be a polynomial ring (Theorem 5.8 and Theorem 5.10).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. O.M. Adamovich, E.O. Golovina: Simple Linear Lie Groups having a Free algebra of Invariants. Questions of Group Theory and Homological Algebra Vyp 2, Yaroslav. Gross. Unir. Yaroslavl' 3–41 (1980) English Translation in Sel. Math. Sov.3 183–220 (1983/84)

  2. N. Bourbaki: Groupes et Algèbres de Lie Chapitres 4, 5 et 6. Hermann Paris 1968

    Google Scholar 

  3. C. Chevalley: Invariants of Finite Groups Generated by Reflections. Amer. Jour. Math. 778–782 (1955)

  4. P. Gordan: Invariantentheorie. Chelsea Publishing Company New York 1987

    MATH  Google Scholar 

  5. V.G. Kac, V.L. Popov, E.B. Vinberg: Sur les groupes lineaires algebriques dont l'algebra des invariantes est libre. C.R. Acad. Sci. Paris Ser. I Math.283 865–878 (1976)

    MathSciNet  Google Scholar 

  6. G. Kempf: Computing Invariants. S. S. Koh (Ed.) Invariant Theory (Lect. Notes Math.1278) 81–94 Springer-Verlag Berlin Heidelberg New York 1987

    Chapter  Google Scholar 

  7. H. Kraft: Geometrische Methoden in der Invariantentheorie (Aspekte der Mathematik D1) Braunschweig-Wiesbaden: Vieweg Verlag 1984

    MATH  Google Scholar 

  8. H. Kraft: ℂ* on affine space. A. Connes, M. Duflo, A. Joseph, R. Rentschler (Eds.) Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Progress in Mathematics92) 561–579 Basel Boston: Birkhäuser Verlag 1990

    Google Scholar 

  9. P. Littelmann: Koreguläre und äquidimensionale Darstellungen. J. Algebra123 No. 1 193–222 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. T. Oda: Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties. New York: Springer Verlag 1988

    MATH  Google Scholar 

  11. V.L. Popov: Modern Developments in Invariant Theory. Proc. Int. Cong. Math. 394–405 (1986)

  12. G.W. Schwarz: Representations of Simple Lie Groups with Regular Rings on Invariants. Inv. Math.49 167–191 (1978)

    Article  MATH  Google Scholar 

  13. G.C. Shephard, J.A. Todd: Finite Unitary Reflection Groups. Can. Jour. Math.6 274–304 (1979)

    MathSciNet  Google Scholar 

  14. R. P. Stanley: Combinatorics and Commutative Algebra. (Progress in Mathematics; 41) Boston Basel Stuttgart: Birkhäuser 1983

    MATH  Google Scholar 

  15. D. Wehlau: A Proof of the Popov Conjecture for Tori. Proc. Amer. Math. Soc.114 839–845 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. D. Wehlau: Constructive Invariant Theory for Tori. Annales De L'Institut Fourier43 (1993) (to appear)

Download references

Author information

Authors and Affiliations

  1. The Royal Military College of Canada, K7K 5L0, Kingston, Ontario, Canada

    David L. Wehlau

Authors
  1. David L. Wehlau
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported in part by NSERC Grant OGP 137522

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wehlau, D.L. When is a ring of torus invariants a polynomial ring?. Manuscripta Math 82, 161–170 (1994). https://doi.org/10.1007/BF02567695

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02567695

Keywords

Search

Navigation