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

Michel Planat; Hishamuddin Zainuddin

doi:10.20944/preprints201609.0067.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 19 September 2016 / Approved: 20 September 2016 / Online: 20 September 2016 (11:14:10 CEST)

Every finite simple group P can be generated by two of its elements. Pairs of generators forP 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 configurationsdefined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All the defined geometries G's have a contextuality parameterclose to its maximal value 1.

finite groups; dessins d’enfants; finite geometries; quantum commutation; quantum contextuality

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

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