A survey on association schemes on triples

TL;DR

This survey reviews the development, properties, and open questions of association schemes on triples (ASTs), a ternary generalization of classical association schemes, highlighting their algebraic structures and construction methods.

Contribution

It provides a comprehensive overview of ASTs, introduces new concepts like fusion and fission ASTs, and discusses examples derived from group actions and combinatorial designs.

Findings

ASTs generalize classical association schemes to ternary relations

Properties of ternary adjacency algebras are characterized

Examples include ASTs from affine special linear groups

Abstract

Association schemes on triples (ASTs) are ternary analogues of classical association schemes, whose relations and adjacency algebras are ternary instead of binary. We provide a survey of the current progress in the study of ASTs, highlighting open questions, suggesting research directions, and producing some related results. We review properties of the ternary adjacency algebras of ASTs, ASTs whose relations are invariant under some group action, and ASTs obtained from 2-designs and two-graphs. We also provide a notion of fusion and fission ASTs, using the AST obtained from the affine special linear group $ASL(2,q)$ as an example.