Circulant association schemes on triples

Abstract

Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras)
are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a
three-dimensional generalisation of association schemes which they called an {\em association scheme on triples} (AST)
and constructed examples of several families of ASTs. Many of their examples used 2-transitive permutation groups:
the non-trivial ternary relations of the ASTs were sets of ordered triples of pairwise distinct points of the
underlying set left invariant by the group; and the given permutation group was a subgroup of automorphisms of
the AST. In this paper, we consider ASTs that do not necessarily admit 2-transitive groups as automorphism
groups but instead a transitive cyclic subgroup of the symmetric group acts as automorphisms. Such ASTs are
called {\em circulant} ASTs and the corresponding ternary relations are called {\em circulant relations}.
We give a complete characterisation of circulant ASTs in terms of AST-regular partitions of the underlying
set. We also show that a special type of circulant, that we call a {\em thin circulant}, plays a key role in
describing the structure of circulant ASTs. We outline several open questions.