There is one last thing about generalised quadrangles that was left remaining after Michael’s excellent series. There is a process which begins with a certain generalised quadrangle arising from a flock, which after a few steps, produces another possibly different generalised quadrangle. I will explain how this works with a classical example.
The classical flock generalised quadrangle
The totally isotropic points and lines of the Hermitian variety (with automorphism group ) is a generalised quadrangle of order . It is the “classical” example of a flock generalised quadrangle, as we can construct it from a linear flock; a partition of the 3-dimensional quadratic cone minus its vertex, into conics whose planes are all incident with a common line. Now we take the point-line dual:
- Point – line dual: We obtain a generalised quadrangle isomorphic to that obtained by taking the points and lines of the elliptic quadric (this can be seen by the beautiful Klein correspondence). This generalised quadrangle is a translation generalised quadrangle (TGQ) and it arises from an egg in the following way. The classical ovoid of , the elliptic quadric , is one of the canonical examples of an egg (the other being an oval of ). Call this egg and consider the following incidence structure. We embed everything into a hyperplane of . Then we take
(i) Affine points of ,
(ii) Hyperplanes of meeting in a tangent plane of
(iii) one object, just the set
(a) lines of meeting in an element of ,
(b) the elements of .
Incidence is just inherited from the ambient space, or is set-theoretic inclusion.
- A TGQ: Then we obtain a TGQ of order , and in the case that is the elliptic quadric of , we end up with . Now we take the translation dual.
- Translation dual: If is odd, the tangent planes form an ovoid of the dual projective space. So we flip the projective space upside down and obtain another egg. In this case, we get another elliptic quadric, but generally, we may obtain something completely different.
- Another TGQ: From the (possibly new) egg, we can construct another translation generalised quadrangle of order , as above. For the classical example, we obtain again.
- Point – line dual: We then take the dual an arrive at a generalised quadrangle of order .
The question that needs to be asked here is when is this generalised quadrangle at the end of this process an elation generalised quadrangle, and when does it arise from a flock?
For the first question, it is not clear to me if it is known if the point-line dual of a TGQ is an EGQ, though I don’t know of any counter-examples. (Could someone in the ether confirm this?). The second question does have an answer.
Good at an element
An egg of is said to be “good at an element” if every ()-space containing and at least two other egg elements, contains exactly egg elements. That is, an egg is induced in every such ()-space on . By a beautiful theorem of Thas (1999), we have
Theorem: An egg of , odd, is good at an element if and only if is the translation dual of the point-line dual of a semifield flock GQ.
A nice proof of this result appears in Lavrauw and Penttila (2001). By a result of Johnson (1987), if a TGQ is the dual of a flock GQ then the flock is a semifield flock. Moreover, Koen Thas proved in 2007 the following as a solution to a question Kantor gave in a conference talk :
Theorem: Let be the point-line dual of a non-classical good TGQ of order with odd. Then there is a unique elation point in , and it admits a unique elation group.
Semifield flock GQs are rare
- The classical example . The point-line dual is isomorphic to its translation dual.
- Arising from the Kantor-Knuth flocks, all odd prime powers . The point-line dual is isomorphic to its translation dual.
- Arising from the Cohen-Ganley flocks, is a power of 3. The translation dual of its point-line dual is a different generalised quadrangle known as a Roman GQ.
- Arising from the Penttila-Williams flock, . The translation dual of its point-line dual is also a different generalised quadrangle.
There are a lot of details, and hopefully I have got them correct, but I welcome any comment on this brief synopsis.