Generalized quadrangle

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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 $p g (s, t,α)$ .

1 Definition
2 Duality
3 Properties
4 Classical generalized quadrangles
5 Non-classical examples
6 Restrictions on parameters
7 References

[edit] Definition

A generalized quadrangle is an incidence structure (P,B,I), with $I\subseteq P\times B$ 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 ( $s\geq 1$ ) 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 ( $t\geq 1$ ) 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.

[edit] 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.

[edit] 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.
$(s + t) | s t (s + 1)(t + 1)$
$s\neq 1 \Longrightarrow t\leq s^2$
$t\neq 1 \Longrightarrow s\leq t^2$

[edit] 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 :

    (this is just a grid)

A hermitian variety $H (n, q 2)$ has projective index 1 if and only if n is 3 or 4. We find :

A symplectic polarity in $P G (2 d + 1, q)$ 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.

[edit] Non-classical examples

Let O be a hyperoval in $P G (2, q)$ with q an even prime power, and embed that projective (desarguesian) plane $π$ into $P G (3, q)$ . Now consider the incidence structure $T_2^{*}(O)$ where the points are all points not in $π$ , the lines are those not on $π$ , intersecting $π$ in a point of O, and the incidence is the natural one. This is a (q-1,q+1)-generalized quadrangle.
Let q be a prime power (odd or even) and consider a symplectic polarity $θ$ in $P G (3, q)$ . Choose a random point p and define $π = p θ$ . Let the lines of our incidence structure be all absolute lines not on $π$ together with all lines through p, and let the points be all points of $P G (3, q)$ except those in $π$ . The incidence is again the natural one. We obtain once again a (q-1,q+1)-generalized quadrangle

[edit] Restrictions on parameters

By using grids and dual grids, any integer $z$ , $z\geq 1$ allows generalized quadrangles with parameters $(1, z)$ and $(z,1)$ . Apart from that, only the following orders have been found possible until now, with $q$ a random 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)

[edit] References

The standard references are:

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