Introduction to generalised quadrangles

For the purposes of classifying “single-bonded” root systems, we would like to consider the following (finite) incidence structure, known as generalised quadrangle

A pair (P,L), with P a set (elements of P are referred to as points) and L\subset 2^P (elements of L are referred to as lines) is called a generalised quadrangle provided the following holds:

  1. the lines are of size at least 2
  2. for every point there is at least one line containing it
  3. for every point p and every line \ell there exists unique line m\ni p such that \ell\cap m\neq\emptyset.

The 3rd axiom implies that two lines intersect in a most one point, and so indeed can be visualised as such. This permits one to talk about collinearity of points: two points are collinear iff there exists a line containing them both. Also, note that it implies that in (P,L) there are no “triangles”, i.e. triples of lines each two of which intersect, so that there are 3 non-collinear intersection points in total.

Examples

  1. a “line” (P,\{P\}), i.e. there is just one line on all the points
  2. a “claw”, i.e. (P,L) such that there exists p\in P such that p\in\ell for all \ell \in L (and so any x\in P-\{p\} is on just one line)
  3. a complete bipartite graph, i.e. (P,L) with P being the set of vertices of a complete bipartite graph and L being the set of its edges (so in particular every line contains just 2 points)
  4. a “grid”, i.e. (P,L) with L being two sets of parallel lines in \mathbb{R}^2 and P being their pairwise intersections
  5. for \Omega=\{1,\dots,6\} let (P,L) with P being the set \binom{\Omega}{2} of unordered pairs of elements of \Omega,b and L the set of partitions of \Omega into three pairs; in particular |P|=|L|=15.

Note that the 3rd and the 4th examples are dual to each other in the following sense.
In (L,P) the points are the lines of (P,L) and the lines are the sets of lines collinear to a point p, for each p\in P.

Let (P,L) be a generalised quadrangle. Then its dual, the incidence system (L,P), is also a generalised quadrangle, provided that it is neither a “line” nor a “claw”.

We have to exclude “lines” and “claws” as their duals have lines of size 1.

To emphasise the roles of points and lines as dual to each other, we say that p\in P and \ell\in L are incident if p\in\ell.
It is an interesting exercise to show that the 5th example is self-dual, i.e. there exists a bijection \phi:P\to L preserving the incidence point-line relation.

Regularity
The 3rd axiom implies that for any two skew (i.e. non-intersecting) lines \ell,\ell' there is a bijection \psi_{\ell,\ell'} between the sets of points incident to them. This means that any two lines \ell,\ell' without \psi_{\ell,\ell'} must intersect. One can show that more than two different cardinalities are only possible for a “claw”, as follows:
Three lines a,b,c with three different cardinalities must have a common intersection point, say p. A line \ell that is not on p can intersect at most one of the three lines, say a. Therefore there are bijections \psi_{b,\ell} and \psi_{\ell,c}, and \psi_{b,\ell}\circ\psi_{\ell,c} is a bijection between points on b and points on c, contradiction proving the claim.

Similarly, one can show that two different cardinalities are only possible for a “grid” or a “claw”. Indeed, let a,b be two skew lines, and c a line of a different cardinality. Then every line c'\neq c that joins a point on a to a point on b must have the same cardinality as c. On the other hand, every line of a cardinality different from the one of a intersects both a and b. In particular these lines do not intersect. Similarly, the lines with the same cardinality as a intersect c and c'. It follows that we have a “grid”.

It is also not hard to see that when all the lines are of size 2 we have the complete bipartite graph example.

From now on let us assume that all the lines has the same cardinality s,and that (P,L) has the dual. Looking at the dual, we either encounter one of the “easy” cases as above, or derive that all the dual lines must have the same cardinality t. Such (P,L) are called regular.

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