Some interesting open problems on hemisystems

Gordon, Michael and I have just submitted our second paper together, this time we produce examples of hemisystems in small flock generalised quadrangles. So what are these things anyway? Generalised quadrangles have been discussed in depth in Michael’s posts (see this), and in particular, there is an exposition there on GQs arising from BLT-sets/flocks/q-clans. Theses are the only known GQs with parameters (q^2,q) where q is odd. A hemisystem of a generalised quadrangle of order (s,t) is a set \mathcal{H} of half of the lines such that every point is on (t+1)/2 lines of \mathcal{H}. These objects are curious in that they give rise to partial quadrangles, Q-antipodal cometric association schemes and strongly regular graphs. We have already written an earlier post containing some of the history of the subject, but to summarise, it was thought for forty years that Segre’s unique example of H(3,3^2) was the only example, Cossidente and Penttila (2005) showed there were infinitely many in the classical GQ H(3,q^2), and then we showed (2010) that every flock GQ has a hemisystem. Apart from these examples, we find some interesting examples in flock GQs of order (q^2,q) where q\le 11.

One of the main results of our paper is that we have classified the hemisystems of flock GQs of order (25,5):

  • H(3,5^2) hemisystems:
    • Cossidente-Penttila hemisystem
    • 3.A7-hemisystem
  • Fisher-Thas-Walker-Kantor-Betten(5) hemisystems:
    • a hemisystem found by Bamberg, De Clerck and Durante
    • two new ones (stabilisers 5^2:(4\times S_3) and S_3).

Along the way, we came upon some interesting problems that are still open…

Problem 1:

For each odd prime power q, there is an irreducible cyclic subgroup of automorphisms of H(3,q^2), which we call a Singer type element. By computer, we found hemisystems invariant under a Singer type element for q\in\{3,5,7,9,11,17,19,23,27\}; none exist for q\in\{13,25\}. Therefore, we might ask…

Does there exist a hemisystem invariant under a Singer type element for all odd prime powers q \not\equiv 1 \pmod{12}?

Problem 2:

Again we look at hemisystems of H(3,q^2). The stabiliser of three non-degenerate points spanning a non-degenerate plane has shape (q+1)^3:(S_3\times 2f). For example we could fix three points of the canonical basis, assuming the nicest Hermitian form, and we would then have diagonal matrices diag(\lambda_1,\lambda_2,\lambda_3,1). Now take those such matrices which have \lambda_1 \lambda_2\lambda_3=1. Then we end up with a group isomorphic to C_{q+1}^2. For all q\le 17, there exists a hemisystem invariant under this group. Strangely, none exists for q\in\{19,23,25,27,29\}.

What is going on here!?

Problem 3:

For q\in\{3,7,11,19\}, there exists a hemisystem of H(3,q^2) invariant under a group isomorphic to 2^4.A_5. So naturally we ask…

Does there exist a hemisystem of H(3,q^2) invariant under a group isomorphic to 2^4.A_5, for all q\equiv 3\pmod{4}?

Notice that this would be a strange family of hemisystems, as they grow with q but their stabiliser remains constant.

Problem 4:

Is there a hemisystem with trivial group?

Problem 5:

Is the full automorphism group of the strongly regular graph obtained from a hemisystem always induced by the stabiliser of the hemisystem in the automorphism group of the generalized quadrangle?

There are some other problems in the paper, which are a bit too technical to state here.


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