Cameron-Liebler k-sets in subspaces and non-existence conditions

TL;DR

This paper extends the classification of Cameron-Liebler sets from lines to k-spaces in projective and affine geometries, establishing new non-existence conditions for certain dimensions and configurations.

Contribution

It generalizes existing concepts to k-spaces and derives strong non-existence conditions for Cameron-Liebler sets in PG(n,q) and AG(n,q).

Findings

Non-existence condition for n ≥ 3k+3 in PG(n,q)

Improved non-existence condition for AG(n,q) with n ≥ 2k+2

Extension of classification results to higher-dimensional subspaces

Abstract

In this article we generalize the concepts that were used in the PhD thesis of Drudge to classify Cameron-Liebler line classes in PG$(n,q), n\geq 3$, to Cameron-Liebler sets of $k$-spaces in PG$(n,q)$ and AG$(n,q)$. In his PhD thesis, Drudge proved that every Cameron-Liebler line class in PG$(n,q)$ intersects every $3$-dimensional subspace in a Cameron-Liebler line class in that subspace. We are using the generalization of this result for sets of $k$-spaces in PG$(n,q)$ and AG$(n,q)$. Together with a basic counting argument this gives a very strong non-existence condition, $n\geq 3k+3$. This condition can also be improved for $k$-sets in AG$(n,q)$, with $n\geq 2k+2$.