Spin models and distance-regular graphs of $q$-Racah type

TL;DR

This paper constructs a spin model from a specific class of distance-regular graphs called $q$-Racah type, explores the algebraic and combinatorial properties of a central element, and discusses the implications and open problems in this framework.

Contribution

It introduces a method to derive a spin model from $q$-Racah type graphs using a central element in the subconstituent algebra, linking algebraic and combinatorial structures.

Findings

Construction of a spin model from $q$-Racah type graphs

Identification of a central element $Z$ in the subconstituent algebra

Reversal of the construction to recover $Z$ from the spin model

Abstract

Let $\Gamma$ denote a distance-regular graph, with vertex set $X$ and diameter $D\geq 3$. We assume that $\Gamma$ is formally self-dual and $q$-Racah type. We also assume that for each $x \in X$ the subconstituent algebra $T=T(x)$ contains a certain central element $Z=Z(x)$. We use $Z$ to construct a spin model $\sf W$ afforded by $\Gamma$. We investigate the combinatorial implications of $Z$. We reverse the logical direction and recover $Z$ from $\sf W$. We finish with some open problems.