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- Title
CIRCULANT ASSOCIATION SCHEMES ON TRIPLES.
- Authors
BHATTACHARYA, PRABIR; PRAEGER, CHERYL E.
- 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 relation, that we call a thin circulant, plays a key role in describing the structure of circulant ASTs. We outline several open questions.
- Subjects
AUTOMORPHISM groups; INVARIANT sets; PERMUTATION groups; POINT set theory; AUTOMORPHISMS; COMBINATORICS
- Publication
New Zealand Journal of Mathematics, 2021, Vol 52, p153
- ISSN
1171-6096
- Publication type
Article
- DOI
10.53733/106