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

Abstract

Let $\Gamma$ denote a distance-regular graph, with vertex set $X$ and diameter $D\ge 3$. We assume that $\Gamma$ is formally self-dual and $q$-Racah type. Let $A$ denote the adjacency matrix of $\Gamma$. Pick $x\in X$, and let $A^{\ast }=A^{\ast }\left(x\right)$ denote the dual adjacency matrix of $\Gamma$ with respect to $x$. The matrices $A,A^{\ast }$ generate the subconstituent algebra $T=T\left(x\right)$. We assume that for every choice of $x$ the algebra $T$ contains a certain central element $Z=Z\left(x\right)$ whose significance is illuminated by the following relations: $A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}=Z,B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}=Z,C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}}=Z.$ The matrices $A$, $B$ satisfy $A=(A-\varepsilon I)/\alpha$ and $B=(A^{\ast }-\varepsilon I)/\alpha$, where $\alpha ,\varepsilon$ are complex scalars used to describe the eigenvalues of $A$ and $A^{\ast }$. The matrix $C$ is defined using the third displayed equation. We use $Z$ to construct a spin model $W$ afforded by $\Gamma$. We investigate the combinatorial implications of $Z$. We reverse the logical direction and recover $Z$ from $W$. We finish with some open problems.