Geodesic
Cameron-Liebler line classes in PG(n, 5)
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 2, pp. 158-172
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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. 
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I. Matkin. Cameron-Liebler line classes in PG(n, 5). Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 2, pp. 158-172. http://geodesic.mathdoc.fr/item/TIMM_2018_24_2_a14/

[1] Cameron P.J., Liebler R. A., “Tactical decompositions and orbits of projective groups”, Linear Algebra Appl., 46 (1982), 91–102 | DOI | MR | Zbl

[2] Bamberg J., Penttila T., “Overgroups of cyclic Sylow subgroups of linear groups”, Comm. Alg., 36:7 (2008), 2503–2543 | DOI | MR | Zbl

[3] Penttila T., “Cameron-Liebler line classes in PG(3,q)”, Geom. Dedicata, 37:3 (1991), 245–252 | DOI | MR | Zbl

[4] Drudge K., “On a conjecture of Cameron and Liebler”, European J. Combin., 20:4 (1999), 263–269 | DOI | MR | Zbl

[5] Bruen A.A., Drudge K., “The construction of Cameron-Liebler line classes in PG(3,q)”, Finite Fields Appl., 5:1 (1999), 35–45 | DOI | MR | Zbl

[6] J. De Beule, J. Demeyer, K. Metsch, M. Rodgers, “A new family of tight sets in $Q^+(5,q)$”, Des. Codes Cryptogr., 78:3 (2016), 655–678 | DOI | MR | Zbl

[7] Feng T., Momihara K., Xiang Q., “Cameron-Liebler line classes with parameter $x = (q^2-1)/2$”, J. Combin. Theory Ser. A, 133 (2015), 307–338 | DOI | MR | Zbl

[8] Gavrilyuk A.L., Matkin I., Penttila T., “Derivation of Cameron-Liebler line classes”, Des. Codes Cryptogr., 86:1 (2018), 231–236 | DOI | MR | Zbl

[9] Gavrilyuk A.L., Metsch K., “A modular equality for Cameron-Liebler line classes”, J. Combin. Theory Ser. A, 127 (2014), 224–242 | DOI | MR | Zbl

[10] Govaerts P., Penttila T., “Cameron-Liebler line classes in PG(3,4)”, Bull. Belg. Math. Soc. Simon Stevin, 12:5 (2006), 793–804 | MR

[11] Rodgers M., “Cameron-Liebler line classes”, Des. Codes Cryptogr., 68:1–3 (2013), 33–37 | DOI | MR | Zbl

[12] Drudge K., Extremal sets in projective and polar spaces, Thesis (Ph.D.), The University of Western Ontario (Canada), 1998, 111 pp. | MR | Zbl

[13] Gavrilyuk A.L., Mogilnykh I.Yu., “Cameron-Liebler line classes in PG(n,4)”, Des. Codes Cryptogr., 73:3 (2014), 969–982 | DOI | MR | Zbl

[14] Gavrilyuk A.L., Matkin I., Cameron-Liebler line classes in PG(3,5), 19 pp., arXiv: https://arxiv.org/pdf/1803.10442.pdf