Association schemes for diagonal groups

Abstract

For any finite group G, and any positive integer n, we construct an association scheme which admits the diagonal group Dn(G) as a group of automorphisms. The rank of the association scheme is the number of partitions of n into at most |G| parts, so is p(n) if |G| ≥ n; its parameters depend only on n and |G|. For n=2, the association scheme is trivial, while for n=3 its relations are the Latin square graph associated with the Cayley table of G and its complement. A transitive permutation group G is said to be AS-free if there is no non-trivial association scheme admitting G as a group of automorphisms. A consequence of our construction is that an AS-free group must be either 2-homogeneous or almost simple. We construct another association scheme, finer than the above scheme if n>3, from the Latin hypercube consisting of n-tuples of elements of G with product the identity.