A generalization of the Askey-Wilson relations using a projective geometry

In this paper, we present a generalization of the Askey-Wilson relations that involves a projective geometry. A projective geometry is defined as follows. Let $h>k\ge 1$ denote integers. Let ${\mathbb{F}}_q$ denote a finite field with $q$ elements. Let $\mathcal{V}$ denote an $(h+k)$-dimensional vector space over ${\mathbb{F}}_q$. Let the set $P$ consist of the subspaces of $\mathcal{V}$. The set $P$, together with the inclusion partial order, is a poset called a projective geometry. We define a matrix $A\in {\text{Mat}}_P(\mathbb{C})$ as follows. For $u,v\in P$, the $(u,v)$-entry of $A$ is $1$ if each of $u,v$ covers $u\cap v$, and $0$ otherwise. Fix $y\in P$ with $\dim y=k$. We define a diagonal matrix $A^{\ast }\in {\text{Mat}}_P(\mathbb{C})$ as follows. For $u\in P$, the $(u,u)$-entry of $A^{\ast }$ is $q^{\dim (u\cap y)}$. We show that \begin{align*} &A^2A^{}-\bigl(q+q^{-1}\bigr)AA^{}A+A^{}A^{2}-$\mathcal{Y}$\bigl(AA^{}+A^{}A\bigr)-$\mathcal{P}$ A^{}=ΩA+G, \newline &A^{2}A-\bigl(q+q^{-1}\bigr) A^AA^+AA^{2}=$\mathcal{Y}$A^{2}+ΩA^{}+G^{}, \end{align} where $\mathcal{Y},\mathcal{P},\mathrm{\Omega },G,G^{\ast }$ are matrices in ${\text{Mat}}_P(\mathbb{C})$ that commute with each of $A,A^{\ast }$. We give precise formulas for $\mathcal{Y},\mathcal{P},\mathrm{\Omega },G,G^{\ast }$.