On Dihedral Configurations and their Coxeter Geometries

Within the theory of homogeneous coherent configurations, the dihedral configurations play the role which is played by the finite dihedral groups in the theory of finite groups. Imitating Tits’ construction of a geometry from a set of subgroups of a given group, we assign a geometry of rank 2 to each dihedral configuration, its ‘Coxeter geometry’. (Each finite generalized polygon is a Coxeter geometry in this sense.) from general results on the relationship between dihedral configurations and their Coxeter geometries, we settle completely the (ordinary) representation theory of the dihedral configurations of rank 7. We obtain three major classes. The Coxeter geometries of the first class are exactly the non-symmetric 2-designs withI=1. The other two classes lead to questions which require a further combinatorial treatment.