In this post I wish to give a construction of some nonclassical generalised quadrangles, that is, ones other than ${\mathsf{H}(3,q^2),\mathsf{H}(4,q^2), \mathsf{W}(3,q),\mathsf{Q}(4,q),\mathsf{Q}^-(5,q)}$ and the dual of ${\mathsf{H}(4,q^2)}$. These were discussed in the previous post.

The construction I will discuss is known as Payne derivation and is due to Stan Payne. It constructs new GQs from old ones.

Regular points

First I need to introduce some terminology and discuss the notion of a regular point. Let ${\mathcal{Q}}$ be a generalised quadrangle of order ${(s,t)}$. Let ${x,y}$ be points of ${\mathcal{Q}}$. We say that ${x}$ is collinear to ${y}$ if there is a line of the GQ containing both ${x}$ and ${y}$. We usually denote this by ${x\sim y}$ and if ${x}$ and ${y}$ are not collinear we write ${x\not\sim y}$. For a set ${S}$ of points let ${S^{\perp}}$ denote the set of points collinear with each element of ${S}$. We say that a point ${x}$ is incident with a line ${\ell}$ if ${\ell}$ contains ${x}$. Again we denote this by ${x\sim \ell}$.

Now suppose that ${x,y}$ are two points such that ${x}$ is not collinear with ${y}$. For each line ${\ell}$ incident with ${x}$, since ${y}$ is not on ${\ell}$ there is a unique point on ${\ell}$ incident with ${y}$. Since ${x}$ lies on ${t+1}$ lines, it follows that ${\{x,y\}^{\perp}}$ has size ${t+1}$. Moreover, for each ${u,v\in\{x,y\}^{\perp}}$ we have that ${u}$ is not incident with ${v}$, otherwise ${x,u,v}$ would be a triangle in the geometry. Thus ${\{u,v\}^\perp}$ has size ${t+1}$. Hence ${\{x,y\}^{\perp\perp}}$ has size at most ${t+1}$. Note that ${x,y\in\{x,y\}^{\perp\perp}}$ so we in fact have that ${2\leq |\{x,y\}^{\perp\perp}|\leq t+1}$. If ${\{x,y\}^{\perp\perp}=t+1}$ for all points ${y}$ not collinear with ${x}$ we say that ${x}$ is a regular point.