Circulant association schemes on triples

TL;DR

This paper characterizes circulant association schemes on triples (ASTs) with cyclic automorphism groups, introducing the concept of thin circulants and providing a complete classification in terms of AST-regular partitions.

Contribution

It provides a complete characterization of circulant ASTs using AST-regular partitions and introduces the notion of thin circulants as a key structural element.

Findings

Complete classification of circulant ASTs

Introduction of thin circulants as a structural concept

Connection between automorphism groups and AST structure

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 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 circulant ASTs and the corresponding ternary relations are called 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 thin circulant, plays a key role in describing the structure of circulant ASTs. We outline several open questions.