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. 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 $\mathsf{W}$ afforded by $\Gamma$. We investigate the combinatorial implications of $Z$. We reverse the logical direction and recover $Z$ from $\mathsf{W}$. We finish with some open problems.

48 pages

References in corpus (3)

• The Universal Askey-Wilson Algebra
• The Askey-Wilson algebra and its avatars
• Askey-Wilson braid algebra and centralizer of $U_q({\mathfrak{sl}}_2)$