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International Conference on Algebraic Informatics

CAI 2015: Algebraic Informatics pp 58–69Cite as

Effective Invariant Theory of Permutation Groups Using Representation Theory

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9270))

Abstract

Using the representation theory of the symmetric group, we propose an algorithm to compute the invariant ring of a permutation group in the non modular case. Our approach has the advantage of reducing the amount of linear algebra computations and exploits a finer combinatorial description of the invariant ring. We build explicit generators for invariant rings by means of the higher Specht polynomials of the symmetric group.

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References

  1. Abdeljaouad, I.: Théorie des Invariants et Applications à la Théorie de Galois effective. Ph.D. thesis, Université Paris 6 (2000)

    Google Scholar 

  2. Bergeron, F.: Algebraic combinatorics and coinvariant spaces. CMS Treatises in Mathematics (2009)

    Google Scholar 

  3. Borie, N.: Calcul des invariants des groupes de permutations par transformée de Fourier. Ph.D. thesis, Laboratoire de Mathématiques, Université Paris Sud (2011)

    Google Scholar 

  4. Colin, A.: Solving a system of algebraic equations with symmetries. J. Pure Appl. Algebra 117/118, 195–215 (1997). algorithms for algebra (Eindhoven, 1996)

    Article  MathSciNet  Google Scholar 

  5. Colin, A.: Théorie des invariants effective; Applications à la théorie de Galois et à la résolution de systèmes algébriques; Implantation en AXIOM. Ph.D. thesis, École polytechnique (1997)

    Google Scholar 

  6. Sage-Combinat community, T.: Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics (2008)

    Google Scholar 

  7. Derksen, H., Kemper, G.: Computational invariant theory. Springer-Verlag, Berlin (2002)

    Book  MATH  Google Scholar 

  8. Faugère, J., Rahmany, S.: Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases. In: Proceedings of the 2009 international symposium on Symbolic and algebraic computation, pp. 151–158 (2009)

    Google Scholar 

  9. The GAP Group: Lehrstuhl D für Mathematik, RWTH Aachen, Germany and SMCS, U. St. Andrews, Scotland: GAP - Groups, Algorithms, and Programming (1997)

    Google Scholar 

  10. Garsia, A., Stanton, D.: Group actions on stanley-reisner rings and invariants of permutation groups. Advances in Mathematics 51(2), 107–201 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gatermann, K.: Symbolic solution of polynomial equation systems with symmetry. Konrad-Zuse-Zentrum für Informationstechnik, Berlin (1990)

    Google Scholar 

  12. Geissler, K., Klüners, J.: Galois group computation for rational polynomials. J. Symbolic Comput. 30(6), 653–674 (2000). algorithmic methods in Galois theory

    Article  MATH  MathSciNet  Google Scholar 

  13. Kemper, G.: The invar package for calculating rings of invariants. IWR Preprint 93–94, University of Heidelberg (1993)

    Google Scholar 

  14. King, S.: Fast Computation of Secondary Invariants (2007). Arxiv preprint math/0701270

  15. Lascoux, A., Schützenberger, M.P.: Polynômes de Schubert. C. R. Acad. Sci. Paris Sér. I Math. 294(13), 447–450 (1982)

    MATH  Google Scholar 

  16. Pouzet, M., Thiéry, N.M.: Invariants algébriques de graphes et reconstruction. C. R. Acad. Sci. Paris Sér. I Math. 333(9), 821–826 (2001)

    Article  MATH  Google Scholar 

  17. Stanley, R.P.: Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.) 1(3), 475–511 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  18. Stein, W., et al.: Sage Mathematics Software (Version 3.3). The Sage Development Team (2009). http://www.sagemath.org

  19. Sturmfels, B.: Algorithms in invariant theory. Springer-Verlag, Vienna (1993)

    Book  MATH  Google Scholar 

  20. Terasoma, T., Yamada, H.: Higher Specht polynomials for the symmetric group. Proc. Japan Acad. Ser. A Math. Sci. 69(2), 41–44 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  21. Thiéry, N.M.: Computing minimal generating sets of invariant rings of permutation groups with SAGBI-Gröbner basis. In: Discrete models (Paris, 2001), pp. 315–328 (electronic) (2001)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire d’Informatique Gaspard Monge, Université Paris Est á Marne-La-Vallée, Champs-sur-marne, France

    Nicolas Borie

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  1. Nicolas Borie
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Correspondence to Nicolas Borie .

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Editors and Affiliations

  1. Institut für Maschinelle, Universität Stuttgart, Stuttgart, Baden-Württemberg, Germany

    Andreas Maletti

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Borie, N. (2015). Effective Invariant Theory of Permutation Groups Using Representation Theory. In: Maletti, A. (eds) Algebraic Informatics. CAI 2015. Lecture Notes in Computer Science(), vol 9270. Springer, Cham. https://doi.org/10.1007/978-3-319-23021-4_6

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  • DOI: https://doi.org/10.1007/978-3-319-23021-4_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-23020-7

  • Online ISBN: 978-3-319-23021-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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