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 of the defined geometries G's have a contextuality parameter close to its maximal value of .one.