Skip to main content
SpringerLink
Log in

Applications of Invariant Theory

  • Chapter
  • First Online:
Computational Invariant Theory

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS))

Abstract

In this chapter we give a survey of some applications of invariant theory. The selection of topics is very incomplete, and so are certainly the references given for each topic. For example, we omit applications to projective geometry, which are very well explained in Sturmfels (Algorithms in invariant theory. Springer-Verlag, Wien, New York, 1993, Chap. 3). We try to present a wide range of applications from different fields, and exemplify the use of invariant theory in each case.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bernd Sturmfels, Algorithms in Invariant Theory, Springer-Verlag, Wien, New York 1993.

    Book  MATH  Google Scholar 

  2. Alejandro Adem, R. James Milgram, Cohomology of Finite Groups, Springer-Verlag, Berlin, Heidelberg, New York 1994.

    Book  MATH  Google Scholar 

  3. Leonard Evens, The Cohomology of Groups, Oxford University Press, Oxford 1991.

    MATH  Google Scholar 

  4. Alejandro Adem, R. James Milgram, Invariants and cohomology of groups, Bol. Soc. Mat. Mex. 37 (1992), 1–25.

    MATH  Google Scholar 

  5. Richard P. Stauduhar, The determination of Galois groups, Math. Comput. 27 (1973), 981–996.

    Google Scholar 

  6. Katharina Geißler, Jürgen Klüners, Galois group computation for rational polynomials, J. Symb. Comput. 30 (2000), 653–674.

    Article  MathSciNet  MATH  Google Scholar 

  7. Yves Eichenlaub, Problèmes effectifs de la théorie de Galois en degrés 8 à 11, Dissertation, Université Bordeaux 1, 1996.

    Google Scholar 

  8. Katharina Geißler, Zur Berechnung von Galoisgruppen, Diplomarbeit, Technische Univerität Berlin, 1997.

    Google Scholar 

  9. Kurt Girstmair, On the computation of resolvents and Galois groups, Manuscr. Math. 43 (1983), 289–307.

    Article  MathSciNet  MATH  Google Scholar 

  10. Claus Fieker, Jürgen Klüners, Galoisgruppen in Magma, Computeralgebra-Rundbrief 43 (2008), 19–20.

    Google Scholar 

  11. Antoine Colin, Formal computation of Galois groups with relative resolvents, in: Gérard Cohen, Marc Giusti, Teo Mora, eds., Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC-11), Lecture Notes in Computer Science 948, pp. 169–182, Springer-Verlag, Berlin 1995.

    Google Scholar 

  12. Moss Sweedler, Using Gröbner bases to determine the algebraic and transcendental nature of field extensions: Return of the killer tag variables, in: Gérard Cohen, Teo Mora, Oscar Moreno, eds., Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer-Verlag, Berlin, Heidelberg, New York 1993.

    Google Scholar 

  13. Clarence Wilkerson, A primer on the Dickson invariants, Amer. Math. Soc. Contemp. Math. Series 19 (1983), 421–434.

    Google Scholar 

  14. Shreeram S. Abhyankar, Galois embeddings for linear groups, Trans. Amer. Math. Soc. 352 (2000), 3881–3912.

    Google Scholar 

  15. Gunter Malle, B. Heinrich Matzat, Inverse Galois Theory, Springer-Verlag, Berlin, Heidelberg 1999.

    Book  MATH  Google Scholar 

  16. Emmy Noether, Gleichungen mit vorgeschriebener Gruppe, Math. Ann. 78 (1918), 221–229.

    Google Scholar 

  17. H. W. Lenstra, Rational functions invariant under a finite abelian group, Invent. Math. 25 (1974), 299–325.

    Google Scholar 

  18. Richard G. Swan, Invariant rational functions and a problem of Steenrod, Invent. Math. 7 (1969), 148–158.

    Google Scholar 

  19. David J. Saltman, Noether’s problem over an algebraically closed field, Invent. Math. 77 (1984), 71–84.

    Google Scholar 

  20. David J. Saltman, Groups acting on fields: Noether’s problem, Contemp. Mathematics 43 (1985), 267–277.

    Google Scholar 

  21. Gregor Kemper, Das Noethersche Problem und generische Polynome, Dissertation, Universität Heidelberg, 1994a, also available as: Preprint 94–49, IWR, Heidelberg, 1994.

    Google Scholar 

  22. Gregor Kemper, Elena Mattig, Generic polynomials with few parameters, J. Symb. Comput. 30 (2000), 843–857.

    Article  MathSciNet  MATH  Google Scholar 

  23. F. Seidelmann, Die Gesamtheit der kubischen und biquadratischen Gleichungen mit Affekt bei beliebigem Rationalitätsbereich, Math. Ann. 78 (1918), 230–233.

    Google Scholar 

  24. Serge Lang, Algebra, Addison-Wesley Publishing Co., Reading, Mass. 1985.

    MATH  Google Scholar 

  25. Gregor Kemper, Gunter Malle, Invariant fields of finite irreducible reflection groups, Math. Ann. 315 (1999), 569–586.

    Google Scholar 

  26. Patrick A. Worfolk, Zeros of equivariant vector fields: Algorithms for an invariant approach, J. Symb. Comput. 17 (1994), 487–511.

    Google Scholar 

  27. Helmer Aslaksen, Shih-Ping Chan, Tor Gulliksen, Invariants of S 4 and the shape of sets of vectors, Appl. Algebra Engrg. Comm. Comput. 7 (1996), 53–57.

    Article  MathSciNet  MATH  Google Scholar 

  28. Nicolas M. Thiéry, PerMuVAR, a library for mupad for computing in invariant rings of permutation groups, http://permuvar.sf.net/.

  29. Nicolas M. Thiéry, Algebraic invariants of graphs; a study based on computer exploration, SIGSAM Bulletin 34 (2000), 9–20.

    Google Scholar 

  30. Nicolas M. Thiéry, Computing minimal generating sets of invariant rings of permutation groups with SAGBI-Gröbner basis, in: International Conference DM-CCG, Discrete Models - Combinatorics, Computation and Geometry, Paris, July 2–5 2001, 2001, to appear.

    Google Scholar 

  31. S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, London 1960.

    Google Scholar 

  32. Maurice Pouzet, Quelques remarques sur les résultats de Tutte concernant le problème de Ulam, Publ. Dép. Math. (Lyon) 14 (1977), 1–8.

    MathSciNet  MATH  Google Scholar 

  33. Nicolas M. Thiéry, Invariants algébriques de graphes et reconstruction; une étude expérimentale, Dissertation, Université Lyon I, Lyon 1999.

    Google Scholar 

  34. Maurice Pouzet, Nicolas M. Thiéry, Invariants algébriques de graphes et reconstruction, Comptes Rendus de l’Académie des Sciences 333 (2001), 821–826.

    Google Scholar 

  35. Richard P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. 1(3) (1979), 475–511.

    Google Scholar 

  36. Richard P. Stanley, Combinatorics and invariant theory, in: Relations Between Combinatorics and Other Parts of Mathematics (Columbus, Ohio 1978), Proc. Symp. Pure Math. 34, pp. 345–355, Am. Math. Soc., Providence, RI 1979.

    Google Scholar 

  37. B. Char, K. Geddes, G. Gonnet, M. Monagan, S. Watt, Maple Reference Manual, Waterloo Maple Publishing, Waterloo, Ontario 1990.

    Google Scholar 

  38. Louis Solomon, Partition identities and invariants of finite groups, J. Comb. Theory, Ser. A 23 (1977), 148–175.

    Google Scholar 

  39. Larry Smith, Polynomial Invariants of Finite Groups, A. K. Peters, Wellesley, Mass. 1995.

    Google Scholar 

  40. A. Elashvili, M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math., New Ser. 9 (1998), 233–238.

    Google Scholar 

  41. A. Elashvili, M. Jibladze, D. Pataraia, Combinatorics of necklaces and “Hermite reciprocity”, J. Algebr. Comb. 10 (1999), 173–188.

    Google Scholar 

  42. N.J.A. Sloane, Error-correcting codes and invariant theory: New applications of a nineteenth-century technique, Amer. Math. Monthly 84 (1977), 82–107.

    Google Scholar 

  43. F. Jessie MacWilliams, A theorem on the distribution of weights in a systematic code, Bell Syst. Tech. J. 42 (1963), 79–84.

    Google Scholar 

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

    Google Scholar 

  45. Andrew M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, in: Actes Congr. internat. Math., 3, 1970, pp. 211–215, Gauthier-Villars, Paris 1971.

    Google Scholar 

  46. Eric M. Rains, N.J.A. Sloane, Self-dual codes, in: Vera S. Pless, ed., Handbook of Coding Theory, vol. 1, pp. 177–294, Elsevier, Amsterdam 1998.

    Google Scholar 

  47. Eiichi Bannai, Steven T. Dougherty, Masaaki Harada, Manabu Oura, Type II codes, even unimodular lattices, and invariant rings, IEEE Trans. Inf. Theory 45 (1999), 1194–1205.

    Google Scholar 

  48. Gabriele Nebe, Eric M. Rains, Neil J. A. Sloane, Self-dual Codes and Invariant Theory, vol. 17 of Algorithms and Computation in Mathematics, Springer-Verlag, Berlin 2006.

    Google Scholar 

  49. D. H. Sattinger, Group Theoretic Methods in Bifurcation Theory, Lecture Notes in Math. 762, Springer-Verlag, Berlin, Heidelberg, New York 1979.

    Google Scholar 

  50. Martin Golubitsky, David G. Schaeffer, Singularities and Groups in Bifurcation Theory I, Applied Mathematical Sciences 51, Springer-Verlag, New York 1985.

    Google Scholar 

  51. Martin Golubitsky, Ian Stewart, David G. Schaeffer, Singularities and Groups in Bifurcation Theory II, Applied Mathematical Sciences 69, Springer, New York 1988.

    Google Scholar 

  52. Karin Gatermann, Computer Algebra Methods for Equivariant Dynamical Systems, Lecture Notes in Mathematics 1728, Springer-Verlag, Berlin, Heidelberg 2000.

    Google Scholar 

  53. Karin Gatermann, F. Guyard, Gröbner bases, invariant theory and equivariant dynamics, J. Symb. Comput. 28 (1999), 275–302.

    Article  MATH  Google Scholar 

  54. Giuseppe Gaeta, Frank D. Grosshans, Jürgen Scheurle, Sebastian Walcher, Reduction and reconstruction for symmetric ordinary differential equations, J. Differential Equations 244 (2008), 1810–1839.

    Google Scholar 

  55. M. V. Jarić, L. Michel, R. T. Sharp, Zeros of covariant vector fields for the point groups: Invariant formulation, J. Physique 45 (1984), 1–27.

    Google Scholar 

  56. Karin Gatermann, Reiner Lauterbach, Automatic classification of normal forms, Nonlinear Analysis, Theory, Methods, and Applications 34 (1998), 157–190.

    Article  MathSciNet  MATH  Google Scholar 

  57. Ali Lari-Lavassani, William F. Langford, Koncay Huseyin, Karin Gatermann, Steady-state mode interactions for D 3 and D 4 -symmetric systems, Dynamics of Continuous, Discrete and Impulsive Systems 6 (1999), 169–209.

    Google Scholar 

  58. Sue Ann Campbell, Philip Holmes, Heteroclinic cycles and modulated travelling waves in a system with D 4 symmetry, Physica D 59 (1992), 52–78.

    Google Scholar 

  59. Michael A. Collins, Keiran C. Thompson, Group theory and the global functional shapes for molecular potential energy surfaces, in: Danail Bonchev, Dennis H. Rouvray, eds., Chemical Group Theory: Techniques and Applications, pp. 191–234, Gordon and Breach Publishers, Reading 1995.

    Google Scholar 

  60. Z. Hashin, Failure criteria for unidirectional fiber composites, J. Appl. Mech. 47 (1980), 329–334.

    Google Scholar 

  61. W. Helisch, Invariantensysteme und Tensorgeneratoren bei Materialtensoren zweiter und vierter Stufe, Dissertation, RWTH Aachen, Aachen 1993.

    Google Scholar 

  62. J. P. Boehler, ed., Applications of Tensor Functions in Solid Mechanics, CISM Courses and Lectures 292, Springer-Verlag, Wien, New York 1987.

    Google Scholar 

  63. G.F. Smith, M.M. Smith, R.S. Rivlin, Integrity bases for a symmetric tensor and a vector. The crystal classes, Arch. Ration. Mech. Anal. 12 (1963), 93–133.

    Article  MathSciNet  MATH  Google Scholar 

  64. Sabine Meckbach, Gregor Kemper, Invariants of textile reinforced composites, preprint, Universität Gh Kassel, Kassel, 1999.

    Google Scholar 

  65. Joseph L. Mundy, Andrew Zisserman, eds., Geometric Invariance in Computer Vision, Artificial Intelligence, MIT Press, Cambridge, MA 1992.

    Google Scholar 

  66. Thomas H. Reiss, Recognizing Planar Objects Using Invariant Image Features, Lecture Notes in Computer Science 676, Springer-Verlag, Berlin 1993.

    Google Scholar 

  67. Kenichi Kanatani, Group-Theoretical Methods in Image Understanding, Springer-Verlag, Berlin, Heidelberg, New York 1990.

    Book  MATH  Google Scholar 

  68. Luc Florack, Image Structure, Computational Imaging and Vision 10, Kluwer, Dordrecht, Boston, London 1997.

    Google Scholar 

  69. Charles A. Rothwell, Andrew Zisserman, David A. Forsyth, Joseph L. Mundy, Fast recognition using algebraic invariants, in: Joseph L. Mundy, Andrew Zisserman, eds., Geometric Invariance in Computer Vision, pp. 398–407, MIT Press, Cambridge, Mass. 1992.

    Google Scholar 

  70. Mireille Boutin, Gregor Kemper, On reconstructing n-point configurations from the distribution of distances or areas, Adv. Applied Math. 32 (2004), 709–735.

    Article  MathSciNet  MATH  Google Scholar 

  71. Gabriel Taubin, David B. Cooper, Object recognition based on moment (or algerbaic) invariants, in: Joseph L. Mundy, Andrew Zisserman, eds., Geometric Invariance in Computer Vision, pp. 375–397, MIT Press, Cambridge, Mass. 1992.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

    Harm Derksen

  2. Zentrum Mathematik - M11, Technische Universität München, Garching, Germany

    Gregor Kemper

Authors
  1. Harm Derksen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gregor Kemper
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Derksen, H., Kemper, G. (2015). Applications of Invariant Theory. In: Computational Invariant Theory. Encyclopaedia of Mathematical Sciences. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48422-7_5

Download citation

Publish with us

Policies and ethics

Search

Navigation