Circulant association schemes on triples




Permutation groups, k-relations, association schemes on triples, circulants


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.  


Download data is not yet available.

Author Biography

Prabir Bhattacharya, Thomas Edison State University

School of Arts, Science & Technology,
Thomas Edison State University,
Trenton, New Jersey, USA




How to Cite

Praeger, C., & Bhattacharya, P. (2021). Circulant association schemes on triples. New Zealand Journal of Mathematics, 52, 153–165.



Vaughan Jones Memorial Special Issue