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

1
Institut FEMTO-ST, CNRS, 15 B Avenue des Montboucons, F-25033 Besançon, France
2
Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Malaysia
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Author to whom correspondence should be addressed.
Academic Editor: Indranil SenGupta
Mathematics 2017, 5(1), 6; https://doi.org/10.3390/math5010006
Received: 19 September 2016 / Revised: 12 December 2016 / Accepted: 5 January 2017 / Published: 14 January 2017
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. View Full-Text
Keywords: finite groups; dessins d’enfants; finite geometries; quantum commutation; quantum contextuality finite groups; dessins d’enfants; finite geometries; quantum commutation; quantum contextuality
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Planat, M.; Zainuddin, H. Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation. Mathematics 2017, 5, 6.

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