SEMINAR: Groups and Combinatorics Seminar: Generalising hemisystems and relative hemisystems

Time and place: 16:00 Friday 26 August in Weatherburn LT

Speaker: Melissa Lee (UWA)

Title: Generalising hemisystems and relative hemisystems

Abstract: Segre determined in 1965 that the only nontrivial m-covers of H(3,q^2), q odd, have m=(q+1)/2. He named these covers hemisystems because they constitute half of the lines on every point. Since then, there have been a variety of generalisations of hemisystems, including to nonclassical generalised quadrangles. Vanhove showed that the dual of a hemisystem, namely a (q+1)/2-ovoid of a dual Hermitian space DH(2d-1,q^2), gives rise to a distance regular graph, and such distance regular graphs would be new for d > 2. Relative hemisystems, defined by Penttila and Williford in 2011, are the analogues of hemisystems for q even. Their definition was motivated by the desire to generate rare primitive Q-polynomial 3-class association schemes. There has been no exploration of relative hemisystems on nonclassical generalised quadrangles in the literature, nor has there been any discussion about relative hemisystems as the unique nontrivial relative m-covers. In this talk, I will explore such generalisations of relative hemisystems, as well as presenting some results on m-ovoids of dual polar spaces, in our search for new distance regular graphs. This talk is a summary of my recently accepted Masters dissertation.