Introduction to generalised quadrangles

By Dima

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:

the lines are of size at least 2

for every point there is at least one line containing it

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

a “line” $(P,\{P\}),$ i.e. there is just one line on all the points
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)
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)
a “grid”, i.e. $(P,L)$ with $L$ being two sets of parallel lines in $\mathbb{R}^2$ and $P$ being their pairwise intersections
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.

Tags: algebra, MAS932, maths

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Equatorial Mathematics

Introduction to generalised quadrangles

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