Separability Number and Schurity Number of Coherent Configurations

To each coherent configuration (scheme) C and positive integer m we associate a natural scheme Ĉ(m) on the m-fold Cartesian product of the point set of C having the same automorphism group as C. Using this construction we define and study two positive integers: the separability number s(C) and the Schurity number t(C) of C. It turns out that s(C) ≤ m iff C is uniquely determined up to isomorphism by the intersection numbers of the scheme Ĉ(m). Similarly, t(C) ≤ m iff the diagonal subscheme of Ĉ(m) is an orbital one. In particular, if C is the scheme of a distance-regular graph Γ, then s(C) = 1 iff Γ is uniquely determined by its parameters whereas t(C) = 1 iff Γ is distance-transitive. We show that if C is a Johnson, Hamming or Grassmann scheme, then s(C) ≤ 2 and t(C) = 1. Moreover, we find the exact values of s(C) and t(C) for the scheme C associated with any distance-regular graph having the same parameters as some Johnson or Hamming graph. In particular, s(C) = t(C) = 2 if C is the scheme of a Doob graph. In addition, we prove that s(C) ≤ 2 and t(C) ≤ 2 for any imprimitive 3/2-homogeneous scheme. Finally, we show that s(C) ≤ 4, whenever C is a cyclotomic scheme on a prime number of points.