Association schemes and permutation groups

A set of zero-one matrices satisfying (CC1)–(CC4) is called a coherent configuration. It is really a combinatorial object, since the conditions on the matrices can be translated into combinatorial conditions on the binary relations Oi. The coherent configuration formed by the orbital matrices of a permutation group G is the orbital configuration of G. Indeed, a coherent configuration is a partition of Ω2 with some additional properties. So the coherent configurations on Ω inherit an order from the lattice of partitions of Ω2. We say that the coherent configuration A is finer than B (or that B is coarser than A) if each relation in A is a subset of some relation in B . The orbital configuration of a permutation group G is clearly the finest coherent configuration on which G acts as a group of strong automorphisms (preserving all the relations Oi).