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Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation
Version 1
: Received: 19 September 2016 / Approved: 20 September 2016 / Online: 20 September 2016 (11:14:10 CEST)
A peer-reviewed article of this Preprint also exists.
Planat, M.; Zainuddin, H. Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation. Mathematics 2017, 5, 6. Planat, M.; Zainuddin, H. Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation. Mathematics 2017, 5, 6.
Abstract
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.
Keywords
finite groups; dessins d’enfants; finite geometries; quantum commutation; quantum contextuality
Subject
Computer Science and Mathematics, Geometry and Topology
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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