Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 20 September 2016 doi:10.20944/preprints201609.0067.v1 Peer-reviewed version available at Mathematics 2017, 5, 6; doi:10.3390/math5010006 Article Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation Michel Planat 1,* and Hishamuddin Zainuddin 2 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, Selangor, Malaysia;
[email protected] * Correspondence:
[email protected]; Tel.: +33-363-082-492 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 the defined geometries G0s have a contextuality parameter close to its maximal value 1. Keywords: finite groups; dessins d’enfants; finite geometries; quantum commutation; quantum contextuality MSC: 81P45, 20D05, 81P13, 11G32, 51E12, 51E30, 20B40 1.