Coherent configurations over copies of association schemes of prime order

Abstract

Let G be a group acting faithfully and transitively on Ωi for Ωi = 1,2. A famous theorem by Burnside implies the following fact: If |Ω1| = |Ω2| 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| = |Ω1| = |Ω2| for some (α, β) ϵ Ω1 × Ω2. 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.