Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation
Institut FEMTO-ST, CNRS, 15 B Avenue des Montboucons, F-25033 Besançon, France
Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Malaysia
Author to whom correspondence should be addressed.
Academic Editor: Indranil SenGupta
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
. It is unusual, but significant to recognize that a
is a Grothendieck’s “dessin d’enfant”
and that a wealth of standard graphs and finite geometries
—such as near polygons and their generalizations—are stabilized by a
. In our paper, tripods
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
have a contextuality parameter close to its maximal value of one.
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. (CC BY 4.0).
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MDPI and ACS Style
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(1):6.
Planat, Michel; Zainuddin, Hishamuddin. 2017. "Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation." Mathematics 5, no. 1: 6.
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