Coherent configurations over copies of association schemes of prime order

Abstract

Let $G$ be a group acting faithfully and transitively on ${\Omega }_i$ for $i=1,2$. A famous theorem by Burnside implies the following fact: If $|{\Omega }_1|=|{\Omega }_2|$ is a prime and the rank of one of the actions is greater than two, then the actions are equivalent, or equivalently $|(\alpha ,\beta {)}^G|=|{\Omega }_1|=|{\Omega }_2|$ for some $(\alpha ,\beta )\in {\Omega }_1\times {\Omega }_2$. In this paper we consider a combinatorial analogue to this fact through the theory of coherent configurations, and give some arithmetic sufficient conditions for a coherent configuration with two homogeneous components of prime order to be uniquely determined by one of the homogeneous components.

arXiv commentTo appear in Ars Mathematica Contemporanea (AMC). The fist author Supported by the Persian Gulf University Research Council