Boolean functions with a geometric domain: a place where algebra and combinatorics meet

Korte inhoud:In 1982 P. Cameron and R. Liebler investigated collineation groups of the finite three dimensional projective space PG(3, q) having the same number of orbits on the lines as on the points. The line orbits of such collineation groups satisfy particular algebraic and geometric properties and are crucial in this classification attempt. These line classes were initially called special line classes and later renamed to Cameron-Liebler line classes in PG(3, q). It was long thought that a classification could be obtained eventually, and it was conjectured that only trivial line classes could exist. A major turn in the investigation arose in 1999 when A. Bruen and K. Drudge disproved this geometrical conjecture, revitalizing the topic. A second important year was 2019, when a generalization of these line classes to sets of k-spaces in PG(n, q) were described. Additionally, a major breakthrough arose in the same year, where these geometrically defined sets of subspaces, i.e. Cameron-Liebler sets, where generalized to Boolean functions satisfying algebraic conditions, i.e. Boolean degree 1 functions. This provided a new insight in existing problems.

In this thesis we will carefully explore Boolean functions derived from geometric objects. Initially, we will focus on the original problem and restrict to degree 1, doing so, we take a step forward towards a classification. This includes the extension and strengthening of known existence conditions for Cameron-Liebler sets of k-spaces by using the underlying algebra. These algebraic techniques also provide a deeper understanding and have shown to be extremely successful. They also provide a sensible way of extending these concepts to affine geometries.

Secondly, we extend our investigation to Boolean functions of general degree over a finite projective space PG(n, q). Many existing results for degree 1 functions may seem naturally generalizable, but this turns out not to be straightforward. More specifically, by mapping out a deep connection between degree t functions and t-designs, we are able to provide strong existence conditions for these functions. Besides, we will also focus on finding interesting examples. Special attention will be given to degree 2, as it is the logical starting point.

Finally, we will translate these valuable algebraic techniques to other geometrical problems. One such example is given by m-ovoids in polar spaces, where we are not only able to extend but also strengthen existing bounds on m. This, in addition with the traditional Cameron-Liebler problems, provide a place where algebra and combinatorics meet.

Jaar van publicatie:2025

Toegankelijkheid:Open