Generalized quadrangle

From Wikipedia, the free encyclopedia

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,α).

Contents

[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 | = (st + 1)(s + 1)
  • | B | = (st + 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\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 :
 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,q2) 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.

[edit] Non-classical examples

  • Let O be a hyperoval in PG(2,q) with q an even prime power, and embed that projective (desarguesian) plane π into PG(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 an integer (odd or even) and consider a symplectic polarity θ in PG(3,q). Choose a random point p and define π = pθ. 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 PG(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 parameters have been found possible until now, with q an arbitrary prime power :

(q,q)
(q,q2) and (q2,q)
(q2,q3) and (q3,q2)
(q − 1,q + 1) and (q + 1,q − 1)

[edit] 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