Characterization of balanced coherent configurations

Let G be a group acting on a finite set Ω. Then G acts on Ω×Ω by its entry-wise action and its orbits form the basis relations of a coherent configuration (or shortly scheme). Our concern is to consider what follows from the assumption that the number of orbits of G on Ωi×Ωj is constant whenever Ωi and Ωj are orbits of G on Ω. One can conclude from the assumption that the actions of G on Ωi's have the same permutation character and are not necessarily equivalent. From this viewpoint one may ask how many inequivalent actions of a given group with the same permutation character there exist. In this article we will approach to this question by a purely combinatorial method in terms of schemes and investigate the following topics: (i) balanced schemes and their central primitive idempotents, (ii) characterization of reduced balanced schemes.