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Invariants of Finite Groups over Finite Fields: Recent Progress and New Conjectures

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Finite Fields with Applications to Coding Theory, Cryptography and Related Areas

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Abstract

Let G be a finite group acting on a polynomial ring A:= F[x1,…, xn] by graded algebra automorphisms. If F is a field of characteristic zero, then due to classical results of Emmy Noether one knows that the invariant ring A G can be generated in degrees less or equal to |G|. If F is a field of positive characteristic p dividing the group order |G|, this is no longer true. The situation in characteristic p not dividing |G| has been clarified recently after being open for several decades. This paper presents an account on these developments, including some related questions and conjectures dealing with constructive and structural properties of modular invariant rings.

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References

  1. Benson D., Polynomial Invariants of Finite Groups, Cambridge Univ. Press 1993.

    Book  MATH  Google Scholar 

  2. Bruns W., Herzog J., Cohen-Macaulay Rings, Cambridge Univ. Press 1993.

    MATH  Google Scholar 

  3. Ellingsrud G., Skjelbred T., Profondeur d’anneaux d’invariants en caractéristique p, Compos. Math. 41, 233–244, (1980).

    MathSciNet  MATH  Google Scholar 

  4. Fleischmann P., Lempken W. On Degree Bounds for Invariant Rings of Finite Groups Over Finite Fields, Cont. Math. 225, 33–41, (1997).

    Google Scholar 

  5. Fleischmann P., Relative trace ideals and Cohen - Macaulay quotients of modular invariant rings in `Computational Methods for Representations of Groups and Algebras’, Dräxler P., Michler G.., Ringel C. M. Ed., Birkhäuser, Basel (1999).

    Google Scholar 

  6. Fleischmann P., The Noether Bound in Invariant Theory of Finite Groups, Adv. in Mathematics, 156, 23–32, (2000).

    Article  MathSciNet  MATH  Google Scholar 

  7. Fleischmann P., Shank R. J., Depth and the Relative Trace Ideal,to appear in Arch. Math.

    Google Scholar 

  8. Fogarty J., On Noether’s Bound for Polynomial Invariants of a Finite Group., Elec. Res. Ann. of the AMS (7), 5–7, (2001).

    MathSciNet  MATH  Google Scholar 

  9. Kemper G., Hughes I., Symmetric Powers of Modular Representations, Hilbert Series and Degree Bounds, Comm. in Algebra 28, 2059–2088, (2000).

    Article  MathSciNet  MATH  Google Scholar 

  10. Kemper G., On the Cohen- Macaulay property of modular invariant rings, J of Algebra 215, 330–351, (1999).

    Article  MathSciNet  MATH  Google Scholar 

  11. Kemper G., The depth of invariant rings and cohomology, with an appendix by K. Magaard, preprint (1999), Universität Heidelberg, to appear in the J. of Algebra.

    Google Scholar 

  12. Lorenz M., Pathak J., On Cohen-Macaulay Rings of Invariants,preprint (2001), Temple University.

    Google Scholar 

  13. Noether E., Der Endlichkeitssatz der Invarianten endlicher Gruppen,Math. Ann. 77, 89–92, (1916).

    Google Scholar 

  14. Noether E., Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charakteristik p, Nachr. Ges. Wiss. Göttingen (1926), 28–35, in ‘Collected Papers’, pp. 485–492, Springer Verlag, Berlin (1983).

    Google Scholar 

  15. Richman D., Explicit generators of the invariants of finite groups, Adv. Math. 124, 49–76, (1996).

    MathSciNet  MATH  Google Scholar 

  16. Richman D., Invariants of finite groups over fields of characteristic p, Adv. Math. 124 (1996), 25–48.

    Article  MathSciNet  MATH  Google Scholar 

  17. Shank R. J., Wehlau D., Noether numbers for sub representations of cyclic groups of prime order,to appear in the Bulletin of the London Mathematical Society.

    Google Scholar 

  18. Landrock P., Finite Group Algebras and their Modules, LMS Lecture Note Series, Cambridge University Press (1984).

    Google Scholar 

  19. Smith L., E. Noether’s bound in the invariant theory of finite groups, Arch. Math. 66, 89–92, (1995).

    Article  Google Scholar 

  20. Smith L., Polynomial Invariants of Finite Groups, A.K. Peters Ltd., (1995).

    MATH  Google Scholar 

  21. Smith L., Polynomial invariants of finite groups, a survey of recent developments, Bull. Amer. Math. Soc. (1997), 211–250.

    Google Scholar 

  22. Smith L., Homological Codimension of Modular Rings of Invariants and the Koszul Complex, preprint (1997).

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury, CT2 7NF, England

    Peter Fleischmann

Authors
  1. Peter Fleischmann
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Editor information

Editors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, 16802, University Park, PA, USA

    Gary L. Mullen

  2. FB6 Mathematik und Informatik, Universität Essen, 45117, Essen, Deutschland

    Henning Stichtenoth

  3. Departamento de Matemáticas, Universidad Autónoma Metropolitana, México, Izatapalapa 09340 Av. San Rafael Atlixco 186, 09340, México, D.F., México

    Horacio Tapia-Recillas

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© 2002 Springer-Verlag Berlin Heidelberg

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Fleischmann, P. (2002). Invariants of Finite Groups over Finite Fields: Recent Progress and New Conjectures. In: Mullen, G.L., Stichtenoth, H., Tapia-Recillas, H. (eds) Finite Fields with Applications to Coding Theory, Cryptography and Related Areas. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59435-9_9

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  • DOI: https://doi.org/10.1007/978-3-642-59435-9_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63976-0

  • Online ISBN: 978-3-642-59435-9

  • eBook Packages: Springer Book Archive

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