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

Michel Planat and Hishamuddin Zainuddin
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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

Mathematics, 2017, vol. 5, issue 1, 1-17

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.

Keywords: finite groups; dessins d’enfants; finite geometries; quantum commutation; quantum contextuality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2017
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