Paper detail

Coherent configurations over copies of association schemes of prime order

Let $G$ be a group acting faithfully and transitively on $Ω_i$ for $i=1,2$. A famous theorem by Burnside implies the following fact: If $|Ω_1|=|Ω_2|$ is a prime and the rank of one of the actions is greater than two, then the actions are equivalent, or equivalently $|(α,β)^G|=|Ω_1|=|Ω_2|$ for some $(α,β)\in Ω_1\times Ω_2$. 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.