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Characterization of Balanced Coherent Configurations

Let $G$ be a group acting on a finite set $Ω$. Then $G$ acts on $Ω\times Ω$ 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 $Ω_i\times Ω_j$ is constant whenever $Ω_i$ and $Ω_j$ are orbits of $G$ on $Ω$. One can conclude from the assumption that the actions of $G$ on ${Ω_i}$'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.