Generalized quadrangle

A generalized quadrangle is an incidence structure. A generalized quadrangle is by definition a polar space of rank two. They are the generalized n-gons with $n=4$. They are also precisely the partial geometries $pg(s,t,alpha)$.

Definition

A generalized quadrangle is an incidence structure $(P,B,I)$, with $Isubseteq\; P\; imes\; B$ an incidence relation, satisfying certain axioms. Elements of $P$ are by definition the points of the generalized quadrangle, elements of $B$ the lines. The axioms are the following:
* There is a $s$ ($sgeq\; 1$) such that on every line there are exactly $s+1$ points. There is at most one point on two distinct lines.
* There is a $t$ ($tgeq\; 1$) such that through every point there are exactly $t+1$ lines. There is at most one line through two distinct points.
* For every point $p$ not on a line $L$, there is a unique line $M$ and a unique point $q$, such that $p$ is on $M$, and $q$ on $M$ and $L$.

$(s,t)$ are the parameters of the generalized quadrangle.

Duality

If $(P,B,I)$ is a generalized quadrangle with parameters '$(s,t)$', then $(B,P,I^\{-1\})$, with $I^\{-1\}$ the inverse incidence relation, is also a generalized quadrangle. This is the dual generalized quadrangle. Its parameters are '$(t,s)$'. Even if $s=t$, the dual structure need not be isomorphic with the original structure.

Properties

* $|P|=(s\; t+1)(s+1)$
* $|B|=(s\; t+1)(t+1)$
* When constructing a graph with as vertices the points of a generalized quadrangle, and with the collinear points connected, one finds a strongly regular graph.
* $(s+t)|st(s+1)(t+1)$
* $s\; eq\; 1\; Longrightarrow\; tleq\; s^2$
* $t\; eq\; 1\; Longrightarrow\; sleq\; t^2$

Classical generalized quadrangles

When looking at the different cases for polar spaces of rank at least three, and extrapolating them to rank 2, one finds these (finite) generalized quadrangles :

* A hyperbolic quadric $Q(3,q)$, a parabolic quadric $Q(4,q)$ and an elliptic quadric $Q(5,q)$ are the only possible quadrics in projective spaces over finite fields with projective index 1. We find these parameters respectively : $Q(3,q)\; :\; s=q,t=1$ (this is just a grid) $Q(4,q)\; :\; s=q,t=q$ $Q(5,q)\; :\; s=q,t=q^2$
* A hermitian variety $H(n,q^2)$ has projective index 1 if and only if n is 3 or 4. We find : $H(3,q^2)\; :\; s=q^2,t=q$ $H(4,q^2)\; :\; s=q^2,t=q^3$
* A symplectic polarity in $PG(2d+1,q)$ has a maximal isotropic subspace of dimension 1 if and only if $d=3$. Here, we find $s=q,t=q$.

The generalized quadrangle derived from $Q(4,q)$ is always isomorphic with the dual of the last structure.

Non-classical examples

* Let "O" be a hyperoval in $PG(2,q)$ with "q" an even prime power, and embed that projective (desarguesian) plane $pi$ into $PG(3,q)$. Now consider the incidence structure $T\_2^\{*\}(O)$ where the points are all points not in $pi$, the lines are those not on $pi$, intersecting $pi$ in a point of "O", and the incidence is the natural one. This is a "(q-1,q+1)"-generalized quadrangle.
* Let "q" be an integer (odd or even) and consider a symplectic polarity $heta$ in $PG(3,q)$. Choose a random point "p" and define $pi=p^\{\; heta\}$. Let the lines of our incidence structure be all absolute lines not on $pi$ together with all lines through "p", and let the points be all points of $PG(3,q)$ except those in $pi$. The incidence is again the natural one. We obtain once again a "(q-1,q+1)"-generalized quadrangle

Restrictions on parameters

By using grids and dual grids, any integer $z$, $zgeq\; 1$ allows generalized quadrangles with parameters $(1,z)$ and $(z,1)$. Apart from that, only the following parameters have been found possible until now, with $q$ an arbitrary prime power :

: $(q,q)$: $(q,q^2)$ and $(q^2,q)$: $(q^2,q^3)$ and $(q^3,q^2)$: $(q-1,q+1)$ and $(q+1,q-1)$

References

* S. E. Payne and J. A. Thas. Finite generalized quadrangles. Research Notes in Mathematics, 110. Pitman (Advanced Publishing Program), Boston, MA, 1984. vi+312 pp. ISBN 0-273-08655-3
* Koen Thas. Symmetry in finite generalized quadrangles. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2004. xxii+214 pp. ISBN 3-7643-6158-1