Skip to main content
Log in

The finite irreducible linear groups with polynomial ring of invariants

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

We prove the following result: LetG be a finite irreducible linear group. Then the ring of invariants ofG is a polynomial ring if and only ifG is generated by pseudoreflections and the pointwise stabilizer inG of any nontrivial subspace has a polynomial ring of invariants. This is well-known in characteristic zero. For the case of positive characteristic we use the classification of finite irreducible groups generated by pseudoreflections due to Kantor, Wagner, Zalesskiî and Serežkin. This allows us to obtain a complete list of all irreducible linear groups with a polynomial ring of invariants.

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

Access this article

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. D. J. Benson,Polynomial Invariants of Finite Groups, LMS Lecture Notes Series, vol. 190, Cambridge University Press, Cambridge,1993.

    Google Scholar 

  2. N. Bourbaki,Groupes et Algèbres de Lie, IV, V, VI, Hermann, Paris, 1968.

    Google Scholar 

  3. D. Carlisle and P. H. Kropholler,Rational invariants of certain orthogonal and unitary groups, Bull. London Math. Soc.24 (1992), 57–60.

    Google Scholar 

  4. D. Carlisle and P.H. Kropholler,Modular invariants of finite symplectic groups, preprint (1992).

  5. C. Chevalley,Invariants of finite groups generated by reflections, Amer. J. Math77 (1955), 778–782.

    Google Scholar 

  6. A. M. Cohen,Finite complex reflection groups, Ann. Scient. Éc. Norm. Sup.9 (1976), 379–436.

    Google Scholar 

  7. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson,Atlas of Finite Groups, Clarendon Press, Oxford, 1985.

    Google Scholar 

  8. M. Demazure,Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math.21 (1973), 287–301.

    Google Scholar 

  9. L. E. Dickson,A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc.12 (1911), 75–98.

    Google Scholar 

  10. G.-M. Greuel, G. Pfister, H. Schönemann, SINGULAR,a Computer Algebra System for Singularity Theory, Algebraic Geometry and Commutative A Algebra, Universität Kaiserslautern, Germany.

  11. V. Kac, K. Watanabe,Finite linear groups whose ring of invariants is a complete intersection, Bull. Amer. Math. Soc.6 (1982), 221–223.

    Google Scholar 

  12. W. Kantor,Subgroups of classical groups generated by long root elements, Trans. Amer. Math. Soc.248 (1979), 347–379.

    Google Scholar 

  13. G. Kemper,A constructive approach to Noether's problem, Manuscripta Math.90 (1996), 343–363.

    Google Scholar 

  14. G. Kemper,Calulating invariant rings of finite groups over arbitrary fields, J. Symbolic Computation21 (1996), 351–366.

    Google Scholar 

  15. H. Nakajima,Invariants of finite groups generated by pseudoreflections in positive characteristic, Tsukuba J. Math.3 (1979), 109–122.

    Google Scholar 

  16. M. Schönert (editor), GAP—Groups, Algorithms, and Programming, fourth ed., Lehrstuhl D für Mathematik, RWTH Aachen, Germany, 1994.

    Google Scholar 

  17. J.-P. Serre,Groupes finis d'automorphismes d'anneaux locaux réguliers, Colloque d'Algèbre ENSJF, Paris, 1967, pp. 8-01–8-11.

  18. G. C. Shephard and J. A. Todd,Finite unitary reflection groups, Canad. J. Math.6 (1954), 274–304.

    Google Scholar 

  19. L. Smith,Polynomial Invariants of Finite Groups, AK Peters, Wellesley, 1995.

    Google Scholar 

  20. R. Steinberg,Differential equations invariant under finite reflection groups, Trans. Math. Soc.112 (1964), 392–400.

    Google Scholar 

  21. A. Wagner,Groups generated by elations, Abh. Math. Seminar Univ. Hamburg41 (1974), 190–205.

    Google Scholar 

  22. A. Wagner,Collineation groups generated by homologies of order greater than 2, Geom. Dedicata7 (1978), 387–398.

    Google Scholar 

  23. A. Wagner,Determination of the finite primitive reflection groups over an arbitrary field of characteristic not 2, I, Geom. Dedicata9 (1980), 239–253; II, ibid. Geom. Dedicata10, 183–189; III, Geom. Dedicata ibid.10, 475–523.

    Google Scholar 

  24. C. Xu,Computing invariant polynomials of p-adic reflection groups, Proc. Symp. Appl. Math., vol. 48, Amer. Math. Soc., Providence, 1994.

    Google Scholar 

  25. A. E. Zalesskiî and V. N. Serežkin,Linear groups generated by transvections, Izv. Akad. Nauk SSSR, Ser. Mat.40 (1976), no. 1, 26–49 (in Russian). English translation: A. E. Zalesskiî and V. N. Serežkin,Linear groups generated by transvections, Math. USSR Izv.10 (1976), 25–46.

    Google Scholar 

  26. A. E. Zalesskiî and V. N. Serežkin,Finite linear groups generated by reflections, Izv. Akad. Nauk SSSR, Ser. Mat.44 (1980), no. 6, 1279–1307 (in Russian). English translation: A. E. Zalesskiî and V. N. Serežkin,Finite linear groups generated by reflections, Math. USSR Izv.17 (1981), 477–503.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

The second author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kemper, G., Malle, G. The finite irreducible linear groups with polynomial ring of invariants. Transformation Groups 2, 57–89 (1997). https://doi.org/10.1007/BF01234631

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01234631

Keywords

Navigation