In this post I wish to give a construction of some nonclassical generalised quadrangles, that is, ones other than and the dual of . These were discussed in the previous post.
The construction I will discuss is known as Payne derivation and is due to Stan Payne. It constructs new GQs from old ones.
First I need to introduce some terminology and discuss the notion of a regular point. Let be a generalised quadrangle of order . Let be points of . We say that is collinear to if there is a line of the GQ containing both and . We usually denote this by and if and are not collinear we write . For a set of points let denote the set of points collinear with each element of . We say that a point is incident with a line if contains . Again we denote this by .
Now suppose that are two points such that is not collinear with . For each line incident with , since is not on there is a unique point on incident with . Since lies on lines, it follows that has size . Moreover, for each we have that is not incident with , otherwise would be a triangle in the geometry. Thus has size . Hence has size at most . Note that so we in fact have that . If for all points not collinear with we say that is a regular point.
I will now look at what actually happens in some of the classical GQs. In , a GQ of order , let , , and . (I will use the alternating form described in the previous post.) Then . Moreover, while Thus and . This 2-space contains points and so the upper bound can be met. Since the automorphism group of is , which has rank 3 on the points of the GQ, it is transitive on pairs of noncollinear points and so we have has size for all pairs of noncollinear points and . Thus all points of are regular.
In , for odd, we can choose a basis of the underlying 5-dimensional vector space such that the quadratic form evaluates to 0 on each and but . Moreover, while . Then and and . The totally singular points in this 3-space are and , and so in the GQ, has size 2. Thus the lower bound can hold. In fact, since the automorphism group of this GQ has rank 3 we have has size 2 for all pairs of noncollinear points.
In , a GQ of order , we can choose a basis such that and . Then . The only totally isotropic points in this two space are and where . There are such values of and so in the GQ, has size .
Let be a generalised quadrangle of order . Given a regular point , we can construct a new generalised quadrangle whose points are the points of not collinear with . The lines come in two flavours:
- (i) the lines of not incident with (if you like to think of the lines as subsets of points then you need to remove any points collinear with from such lines),
- (ii) the sets where is not collinear with . (again, need to remove if we wish to think of subsets of points.)
Incidence is then the incidence inherited from .
This new geometry is a generalised quadrangle. Those interested in a proof of this should consult the original paper `Nonisomorphic generalised quadrangles’ of Payne or the book `Finite generalised quadrangles’ by Payne and Thas.
The lines of the first type are incident with points of and by the GQ property, exactly one of these points is incident with . Hence lines of the first type are incident with points. Also, as is a regular point, each line contains points of . The only such point incident with is itself so lines of the second type also are incident with points.
Given a point of , it is incident with lines of the orginal GQ and as is not collinear with none of these lines contains and so are all lines of . Moreover, is incident with the line . If then and so . Hence is the unique line of the second type containing . Thus each point of is incident with lines. So our new GQ has order .
Now for what generalised quadrangles can we use this construction?
So far the only GQs of order that we have seen are and . We saw above that all points of are regular but none of the points of are regular. Applying Payne derivation to we obtain a generalised quadrangle of order . These GQs has been previously obtained by an alternative construction by Ahrens and Szekeres, and by Marshall Hall in the case where is even.
Remember that we also obtain a GQ of order from the dual.
Now for we obtain a generalised quadrangle of order . We have already seen the generalised quadrangle which also has order . In fact these two GQs are isomorphic.
For , the full automorphism group of our new GQ of order is , which acts transitively on both the points and lines of the GQ.
For it was shown by Grunhöfer, Joswig and Stroppel that the full automorphism group of the GQ of order obtained by Payne derivation from using the regular point is just , that is the stabiliser of in the automorphism group of the original GQ. This group is transitive on points but will have two orbits on the lines corresponding to the two types.
There are other GQs to which we can apply Payne derivation but we haven’t introduced them yet and so they will have to be the subject of another post.