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

TL;DR

This paper extends the Askey-Wilson relations by incorporating projective geometry, defining new matrices based on subspace relations over finite fields, and deriving generalized algebraic relations involving these matrices.

Contribution

It introduces a novel generalization of Askey-Wilson relations using projective geometry and explicitly formulates the associated matrix relations and their formulas.

Findings

Derived new algebraic relations involving matrices from projective geometry.

Provided explicit formulas for matrices in the generalized relations.

Extended the framework of Askey-Wilson relations to a geometric setting.

Abstract

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 $h>k\geq 1$ 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 $A^*\in \text{Mat}_{P}(\mathbb{C})$ as follows. For $u\in P$, the $(u,u)$-entry of $A^{*}$ 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^{*}=\Omega A+G, \newline &A^{*2}A-\bigl(q+q^{-1}\bigr) A^*AA^*+AA^{*2}=\mathcal{Y}A^{*2}+\Omega A^{*}+G^{*}, \end{align*} where $\mathcal{Y}, \mathcal{P}, \Omega, G, G^*$ are matrices in $\text{Mat}_{P}(\mathbb{C})$ that commute with each of $A, A^*$. We give precise formulas for $\mathcal{Y}, \mathcal{P}, \Omega, G, G^*$.