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Abstract
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Let G be a group acting on a finite set ?. Then G acts on ? × ?
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 ?i × ?j is constant whenever ?i and ?j are
orbits of G on ?. One can conclude from the assumption that the
actions of G on ?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.
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