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

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 G s have a contextuality parameter close to its maximal value 1.