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Cameron-Liebler line classes in PG(n, 5)
I. Matkin Chelyabinsk State University
Abstract:
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.
UDC:
514.146
MSC: 51E20,
05B25 Received: 16.02.2018
DOI:
10.21538/0134-4889-2018-24-2-158-172