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Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation

Author

Listed:
  • Michel Planat

    (Institut FEMTO-ST, CNRS, 15 B Avenue des Montboucons, F-25033 Besançon, France)

  • Hishamuddin Zainuddin

    (Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Malaysia)

Abstract

Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not necessarily minimal) permutation representations P . It is unusual, but significant to recognize that a P is a Grothendieck’s “dessin d’enfant” D and that a wealth of standard graphs and finite geometries G —such as near polygons and their generalizations—are stabilized by a D . In our paper, tripods P − D − G of rank larger than two, corresponding to simple groups, are organized into classes, e.g., symplectic, unitary, sporadic, etc. (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All of the defined geometries G ′ s have a contextuality parameter close to its maximal value of one.

Suggested Citation

  • Michel Planat & Hishamuddin Zainuddin, 2017. "Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation," Mathematics, MDPI, vol. 5(1), pages 1-17, January.
    • Handle: RePEc:gam:jmathe:v:5:y:2017:i:1:p:6-:d:87879
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    References listed on IDEAS

    as
    1. Michel Planat, 2015. "A Moonshine Dialogue in Mathematical Physics," Mathematics, MDPI, vol. 3(3), pages 1-12, August.
    2. Mark Howard & Joel Wallman & Victor Veitch & Joseph Emerson, 2014. "Contextuality supplies the ‘magic’ for quantum computation," Nature, Nature, vol. 510(7505), pages 351-355, June.
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