Semi-Invariants, equivariants and algorithms

  • Karin Gatermann


The results from Invariant Theory and the results for semi-invariants and equivariants are summarized in a suitable way for combining with Gröbner basis computation. An algorithm for the determination of fundamental equivariants using projections and a Poincaré series is described. Secondly, an algorithm is given for the representation of an equivariant in terms of the fundamental equivariants. Several ways for the exact determination of zeros of equivariant systems are discussed.


Invariant theory Linear representations of finite groups Gröbner bases Computation of fundamental equivariants Solution of systems of polynomial equations 


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  1. 1.
    Barany, E., Dellnitz, M., Golubitsky, M.: Detecting the Symmetry of Attractors. Physica D67, 66–87 (1993)Google Scholar
  2. 2.
    Becker, Th., Weispenning, V.: Gröbner Bases. A Computational Approach to Commutative Algebra. In: Cooperation with H. Kredel. Berlin, Heidelberg, New York: Springer 1993Google Scholar
  3. 3.
    Chevalley, C.: Invariants of finite groups generated by reflections. Am. J. Math.77, 778–782 (1955)Google Scholar
  4. 4.
    Cox, D., Little, J., O'Shea, D.: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Berlin, Heidelberg, New York: Springer 1992Google Scholar
  5. 5.
    Dieudorme, J., Carrel, J. B.: Invariant Theory, Old and New. New York: Academic Press 1971Google Scholar
  6. 6.
    Fässler, A., Stiefel, E.: Group Theoretical Methods and Their Applications. Boston Birkhäuser 1992Google Scholar
  7. 7.
    Field, M. J., Richardson, R. W.: Symmetry Breaking and the Maximal Isotropy Subgroup Conjecture for Reflection Groups. In: Truesdel, C. (ed) Archive for Rational Mechanics and Analysis Vol. 105. Berlin, Heidelberg, New York: Springer 1989Google Scholar
  8. 8.
    Gatermann, K.: Symbolic solution of polynomial equation systems with symmetry. In: Watanabe, Sh.: Nagata, M. (eds) Proceedings of ISSAC-90 (Tokyo, Japan, August 20–24, 1990), pp 112–119. New York: ACM 1990Google Scholar
  9. 9.
    Gaterman, K.: Werner, B.: Secondary Hopf Bifurcation Caused by Steady-state Steady-state Mode interaction. In: Chaddam, J., Golubitsky, M., Langford, W., Wetton, B. (eds) Pattern Formations: Symmetry Methods and Applications, Fields Institute Com. Series, 1994Google Scholar
  10. 10.
    Gatermann, K.: A remark on the detection of symmetry of attractors. In: Chossat, P. (ed) Dynamics, Bifurcation and Symmetry. Kluwer Academic Publishers, Dordrecht, 1994Google Scholar
  11. 11.
    Golubitsky, M., Stewart, I., Schaeffer, D. G.: Singularities and Groups in Bifurcation Theory. Vol. II, Berlin, Heidelberg, New York: Springer 1988Google Scholar
  12. 12.
    Hearn, A. C.: REDUCE User's Manual, Version 3.5. The RAND Corp., Santa Monica, USA 1993Google Scholar
  13. 13.
    Hochster, M., Eagon, J. A.: Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci. Am. J. Math.93, 1020–1058 (1971)Google Scholar
  14. 14.
    Hochster, M., Roberts, J.: Rings of invariants of reductive groups acting on regular rings are Cohen-Macauley. Adv. Math.13, 115–175 (1974)Google Scholar
  15. 15.
    Jaric, M. V., Michel, L., Sharp, R. T.: Zeros of covariant vector fields for the point groups: invariant formulation. J. Phys.45, 1–27 (1984)Google Scholar
  16. 16.
    Kemper, G. The Invar Package for Calculating Rings of Invariants. Maple Share library, 1993Google Scholar
  17. 17.
    McShane, J. M., Grove, L. C.: Polynomial Invariants of Finite Groups. In Algebras, Groups and Geometries10, 1–12 (1993)Google Scholar
  18. 18.
    Melenk, H., Möller, H. M., Neun, W.: GROEBNER A Package for Calculating Groebner Bases. Available with REDUCE 1992Google Scholar
  19. 19.
    Melenk, H., Möller, H. M., Neun, W.: Symbolic solution of large stationary chemical kinetics problems. Impact. Comput. Sci. Eng.1, 138–167 (1989)Google Scholar
  20. 20.
    Molien, T.: Über die Invarianten der linearen Substitutionsgruppe. Königl. Preuss. Akad. Wiss,pp. 1152–1156, 1897Google Scholar
  21. 21.
    Noether, E.: Der Endlichkeitssatz der Invarianten endlicher Gruppen. Math. Ann.77, 89–92 (1916)Google Scholar
  22. 22.
    Sattinger, D. H.: Group Theoretic Methods in Bifurcation Theory. Lecture Notes in Mathematics vol. 762. Berlin, Heidelberg, New York: Springer 1979Google Scholar
  23. 23.
    Schwarz, G. W.: Lifting Smooth Homotopies of Orbit Spaces. In Institut des Hautes Ètudes Scientifiques, Publications Mathématiques 51, 1980Google Scholar
  24. 24.
    Serre, J. P.: Linear Representations of Finite Groups. Berlin, Heidelberg, New York: Springer 1977Google Scholar
  25. 25.
    Sloane, N. J. A.: Error-correcting codes and invariant theory: new applications of a nineteenthcentury technique. Am. Math. Monthly84, 82–107 (1977)Google Scholar
  26. 26.
    Stanley, R. P.: Invariants of finite groups and their applications to combinatorics. Bulletin Am. Math. Soc.1, 475–511 (1979)Google Scholar
  27. 27.
    Sturmfels, B.: Algorithms in Invariant Theory. Wien: Springer 1993Google Scholar
  28. 28.
    Verscheide, J., Gatermann, K.: Symmetric Newton Polytopes for Solving Sparse Polynomial Systems. Konard-Zuse-Zentrum, Preprint SC 94-3, 1994. Accepted for publication by Advances in Applied MathematicsGoogle Scholar
  29. 29.
    Workfold, P. A.: Zeros of equivariant vector fields: Algorithms for an invariant approach. To appear in J. Symbolic Computation 1994Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Karin Gatermann
    • 1
  1. 1.Konrad-Zuse-Zentrum für Informationstechnik BerlinBerlinWilmersdorf Germany

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