Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation

TL;DR

This paper explores the deep connections between finite simple groups, dessins d'enfants, finite geometries, and quantum contextuality, revealing new geometric structures with potential quantum physical significance.

Contribution

It organizes simple groups into classes via dessins and geometries, and characterizes configurations with high quantum contextuality from small index representations.

Findings

Identified geometries with maximal quantum contextuality parameter

Classified simple groups into geometric and combinatorial classes

Performed exhaustive search for configurations from small index representations

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 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.