Roux schemes which carry association schemes locally

TL;DR

This paper explores roux schemes, a special class of association schemes linked to equiangular lines, characterising their structure, eigenmatrices, and providing new constructions and a uniqueness proof for a notable example.

Contribution

It introduces methods to produce roux matrices from association schemes, characterises roux schemes with specific local properties, and presents new families and a uniqueness result for a key example.

Findings

Roux matrices can be derived from association schemes.

Characterisation of roux schemes via eigenmatrices.

Uniqueness of the Hoggar 64-line configuration in complex space.

Abstract

A roux scheme is an association scheme formed from a special "roux" matrix and the regular permutation representation of an associated group. They were introduced by Iverson and Mixon for their connection to equiangular tight frames and doubly transitive lines. We show how roux matrices can be produced from association schemes and characterise roux schemes for which the neighbourhood of a vertex induces an association scheme possessing the same number of relations as the thin radical. An important example arises from the $64$ equiangular lines in $\mathbb{C}^8$ constructed by Hoggar which we prove is unique (determined by its parameters up to isomorphism). We also characterise roux schemes by their eigenmatrices and provide new families of roux schemes using our construction.