SEMINAR: Groups and Combinatorics Seminar: Perp-systems of Projective Spaces

Groups and Combinatorics Seminar

John Bamberg (UWA)

will speak on

Perp-systems of Projective Spaces

at 1pm on Tuesday 23 November in MLR2.

Abstract: Perp-systems were introduced by De Clerck, Delanote, Hamilton, and Mathon to construct new partial geometries, and hence, possibly new strongly regular graphs. In a projective space PG(d,q) equipped with a non-degenerate polarity "perp", a perp-system is a set of mutually disjoint r-subspaces of PG(d,q) such that every pair of elements from this set are mutually opposite (disjoint from the "perp" of the other), and it must have the maximum theoretically possible size. By “pair” here, we do not require that the two elements be distinct, so in particular, every element of a perp-system is non-singular with respect to "perp". The only known perp-systems are self-polar maximal arcs of PG(2,q), q even, and Mathon’s sporadic example in PG(5,3). Mathon’s example had no geometric construction, it was simply written down in coordinate form, but it was known that its stabiliser was isomorphic to S_5 and that it was a perp-system with respect to a symplectic, hyperbolic and elliptic polarity. We will give an overview of the geometric background necessary to understand perp-systems and present some recent work of De Clerck and the author which provides a geometric construction of Mathon's perp-system.



All welcome.