A duality of scaffolds for translation association schemes

TL;DR

This paper explores a duality property of scaffolds in translation association schemes, confirming a modified conjecture within this class and providing a diagrammatic understanding of their structure.

Contribution

It introduces a modified conjecture on scaffold duality and proves its validity specifically for translation association schemes with abelian automorphism groups.

Findings

Confirmed the modified duality conjecture for translation association schemes.

Provided a diagrammatic interpretation of scaffolds in association schemes.

Enhanced understanding of the structure of translation association schemes.

Abstract

Scaffolds are certain tensors arising in the study of association schemes, and have been (implicitly) understood diagrammatically as digraphs with distinguished "root" nodes and with matrix edge weights, often taken from Bose-Mesner algebras. In this paper, we first present a slight modification of Martin's conjecture (2021) concerning a duality of scaffolds whose digraphs are embedded in a closed disk in the plane with root nodes all lying on the boundary circle, and then show that this modified conjecture holds true if we restrict ourselves to the class of translation association schemes, i.e., those association schemes that admit abelian regular automorphism groups.