I.A. Matkin. Cameron–Liebler line classes in PG(n, 5) ... P. 158-172

A Cameron-Liebler line class with parameter $x$ in a finite projective geometry PG$(n, q)$ of dimension $n$ over a field with $q$ elements is a set $\mathcal{L}$ of lines such that any line $\ell$ intersects $x(q+1)+\chi_{\mathcal{L}}(\ell)(q^{n-1}+\dots+q^2-1)$ lines from $\mathcal{L}$, where $\chi_{\mathcal{L}}$ is the characteristic function of the set $\mathcal{L}$. The generalized Cameron-Liebler conjecture states that for $n>3$ all Cameron-Liebler classes are known and have a trivial structure in some sense (more exactly, up to complement, the empty set, a point-pencil, all lines of a hyperplane, and the union of the last two for nonincident point and hyperplane). The validity of the conjecture was proved earlier by other authors for the cases $q=2$, 3, and 4. In the present paper we describe an approach to proving the conjecture for given $q$ under the assumption that all Cameron-Liebler classes in PG$(3,q)$ are known. We use this approach to prove the generalized Cameron-Liebler conjecture in the case $q=5$.

Keywords: finite projective geometry, Cameron-Liebler line classes.

The paper was received by the Editorial Office on February, 16, 2018.

Funding Agency:

Russian Foundation for Basic Research (Grant Number: 17-301-50004).

Il’ya Aleksandrovich Matkin, Faculty of Mathematics, Chelyabinsk State University, Chelyabinsk, 454001 Russia, e-mail: ilya.matkin@gmail.com.


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