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Splitting Permutation Representations of Finite Groups by Polynomial Algebra Methods

  • Vladimir V. KornyakEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11077)

Abstract

An algorithm for splitting permutation representations of a finite group over fields of characteristic zero into irreducible components is described. The algorithm is based on the fact that the components of the invariant inner product in invariant subspaces are operators of projection into these subspaces. An important part of the algorithm is the solution of systems of quadratic equations. A preliminary implementation of the algorithm splits representations up to dimensions of hundreds of thousands. Examples of computations are given in the appendix.

Notes

Acknowledgments

I am grateful to Yu.A. Blinkov, V.P. Gerdt and R.A. Wilson for fruitful discussions and valuable advice.

References

  1. 1.
    Holt, D.F., Eick, B., O’Brien, E.A.: Handbook of Computational Group Theory. Chapman & Hall/CRC, Boca Raton (2005)Google Scholar
  2. 2.
    Parker, R.: The computer calculation of modular characters (the Meat-Axe). In: Atkinson, M.D. (ed.) Computational Group Theory, pp. 267–274. Academic Press, London (1984)Google Scholar
  3. 3.
    Kornyak, V.V.: Quantum models based on finite groups. J. Phys.: Conf. Ser. 965, 012023 (2018). http://stacks.iop.org/1742-6596/965/i=1/a=012023
  4. 4.
    Kornyak, V.V.: Modeling quantum behavior in the framework of permutation groups. EPJ Web Conf. 173, 01007 (2018).  https://doi.org/10.1051/epjconf/201817301007CrossRefGoogle Scholar
  5. 5.
    Cameron, P.J.: Permutation Groups. Cambridge University Press, Cambridge (1999)Google Scholar
  6. 6.
    Bosma, W., Cannon, J., Playoust, C., Steel, A.: Solving Problems with Magma. University of Sydney. http://magma.maths.usyd.edu.au/magma/pdf/examples.pdf
  7. 7.
    Wilson, R., et al.: Atlas of finite group representations. http://brauer.maths.qmul.ac.uk/Atlas/v3

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Laboratory of Information TechnologiesJoint Institute for Nuclear ResearchDubnaRussia

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