Cameron-Liebler sets of generators in the Klein quadric $Q^+(5,q)$
Jozefien D'haeseleer, Jonathan Mannaert, Leo Storme
TL;DR
The paper studies Cameron-Liebler sets of generators in the Klein quadric $Q^+(5,q)$, aiming to broaden the catalog of nontrivial examples and advance the conjectured structural classification. It leverages the Klein correspondence with PG$(3,q)$ to construct diverse examples from partial line spreads, Baer subgeometries, and scattered linear sets, and to derive characterization results for small parameters $x$. A key contribution is linking Cameron-Liebler sets to holes of maximal partial spreads in PG$(3,q)$, enabling lower bounds and decomposition results that illuminate the geometry of $Q^+(5,q)$. Overall, the work substantially enriches the understanding of Cameron-Liebler sets in the Klein quadric and strengthens connections between finite geometry, polar spaces, and linear-set constructions.
Abstract
We investigate Cameron-Liebler sets of planes in the Klein quadric $Q^+(5,q)$ in PG$(5,q)$. We prove that there are many examples of such Cameron-Liebler sets of planes in the Klein quadric. More specifically, we provide an incomplete list of examples of such Cameron-Liebler sets of planes. By doing so, we also provide some characteristic results regarding these sets in connection with the Klein quadric. These results contribute to an open conjecture posed in [21].