Zoology of atlas-groups: dessins d’enfants, finite geometries and quantum commutation

• Main Authors: Planat, Miche, Zainuddin, Hishamuddin
• Format: Article
• Language: English
• Published: MDPI 2017
• Online Access: http://psasir.upm.edu.my/id/eprint/63744/1/Zoology%20of%20Atlas-Groups%20Dessins%20D%E2%80%99enfants%2C.pdf; http://psasir.upm.edu.my/id/eprint/63744/

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 neccessarily minimal) permutation representations P. It is unusual but significant to recognize that a P is a Grothendieck’s dessin d’enfant D and that most 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 G′s have a contextuality parameter close to its maximal value 1.