Edit: Just realized that John has written his version of this announcement and posted it a short while ago. But we have slightly different viewpoints, so I’ll leave this up anyway!
We’ve just uploaded a new paper to the arxiv entitled “Hemisystems of small flock generalized quadrangles” where we continue (and probably end) our exploration of hemisystems of flock generalized quadrangles. A hemisystem of a GQ (generalized quadrangle) of order is a collection of exactly half the lines of the GQ such that exactly half the lines on each point are in the hemisystem.
The line intersection graph of a GQ is strongly regular, and the subgraph induced by the lines of hemisystem is also strongly regular and is called a partial quadrangle.
Initially, hemisystems were thought to be rather rare – Segre proved that there is a unique example in the classical hermitian generalized quadrangle , and after looking for them quite hard, Jef Thas conjectured that there were no further examples. This conjecture was finally disproved by Cossidente & Penttila who produced an infinite family in .
In our previous paper “Every flock generalized guadrangle has a hemisystem” which has now appeared in the Bulletin of the London Mathematical Society, we gave a general construction showing that every flock generalized quadrangle has a hemisystem, in fact lots of them. This isn’t quite the complete story though, because this construction doesn’t produce all hemisystems, not even all the hemisystems that we knew at the time.
So in the last few months, we’ve been working on finding hemisystems by computer in the small flock generalized quadrangles – where “small” means up to order – basically seeing if the details of the smaller cases can help us say anything in general and seeing if any interesting cases crop up. And the outcome of all this searching is that hemisystems seem to be plentiful – as expected, there are large numbers arising from the general construction, but plenty of others as well.
We did find a couple of interesting examples, primarily in , where the data suggested that an infinite family was lurking, and could perhaps be proved to exist if only we could get just the right way of looking at it. We pushed these families to higher values of to get more data to work with, and found a couple of surprises: One of the candidate families simply stopped when got high enough, and another of the candidate families started to exhibit periodic behaviour, existing for all except for and leading us to conjecture that it only exists when . Figuring out what special properties has when might be a good start to getting a handle on this family.
Sadly, we couldn’t find a proof for any of these families despite quite a bit of effort and so we left them as conjectures. But given that we also found lots more hemisystems in all the flock generalized quadrangles that neither come from the general construction nor have particularly large or interesting automorphism groups, perhaps the main message from the two papers is that hemisystems are not really at all rare, and it’s not worth spending too much time on a single infinite family! We put a few open questions at the end of the paper, but I think it’s fair to say that they are more about filling in the details than fundamental questions about existence.
Given all this, we’ll probably be taking a break from hemisystems now and moving on to some new projects!