Raziskovalni matematični seminar

A tight point set T  in a finite classical polar space is a collection of points with as many pairs of collinear points as possible. To be more precise, the point-collinearity graph of a finite classical polar space P is strongly regular, and the eigenvalues of such a graph give a bound on the number of adjacent vertices in T  in terms of |T|; when this bound is met exactly, we say that T  is a tight set of P . The existence and classification of tight sets gives interesting information about the structure of the polar spaces, and they can also be used to define new strongly regular graphs.

We will specifically be looking at examples of tight sets in the hyperbolic polar spaces Q ⁺(2d-1,q). The tight sets of $ Q ⁺(5,q) have been well studied, since under the Klein correspondence they are equivalent to the study of Cameron-Liebler line classes of PG(3,q). However up to this point, there are no known nontrivial examples of tight sets in Q ⁺(7,q).

In this talk, we describe a method to construct new nontrivial tight sets in Q ⁺(7,q) based on the embedding of a Segre variety S_1,3 in the hyperbolic space, along with some resulting examples. We further look at how this construction can be extended to Q ⁺(2d-1,q) for larger even values of d.

Joint work with Jozefien D'haesseleer.