Circulant association schemes on triples

Cheryl Praeger,Prabir Bhattacharya

doi:10.53733/106

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. 

Highlights

In Definition 2.2 we introduce a special kind of partition of the set X called an ASTregular partition, and we show (Theorem 2.3) that A(I) := {R0, R1, R2, R3} ∪ {RI | I ∈ I} is a circulant association scheme on triples (AST) if I is an AST-regular partition of X
This construction is quite general, and we show that each circulant AST based on Ω arises in this way
Our study of circulant ASTs has raised a number of questions we believe are worthy of further study, and would shed more light on these interesting structures

Summary

Introduction

Several infinite families of ASTs were constructed in [29] It was shown in [29, Theorem 4.1] that, for each 2-transitive permutation group G on Ω, we obtain an AST by taking as the nontrivial ternary relations the G-orbits on the set of triples of pairwise distinct points of Ω. In Definition 2.2 we introduce a special kind of partition of the set X called an ASTregular partition, and we show (Theorem 2.3) that A(I) := {R0, R1, R2, R3} ∪ {RI | I ∈ I} is a circulant AST if I is an AST-regular partition of X This construction is quite general, and we show that each circulant AST based on Ω arises in this way.

AST-Regular Partitions

Thin 3-Circulants

Conclusions and New Problems