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

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.