Generalised quadrangles II

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.

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