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.
REFERENCES
1. Cameron P.J., Liebler R. A. Tactical decompositions and orbits of projective groups. Linear Algebra Appl., 1982, vol. 46, pp. 91–102. doi: 10.1016/0024-3795(82)90029-5 .
2. Bamberg J., Penttila T. Overgroups of cyclic Sylow subgroups of linear groups. Comm. Alg., 2008, vol. 36, no. 7, pp. 2503–2543. doi:10.1080/00927870802070108 .
3. Penttila T. Cameron-Liebler line classes in PG(3,q). Geom. Dedicata, 1991, vol. 37, no. 3, pp. 245–252. doi: 10.1007/BF00181401 .
4. Drudge K. On a conjecture of Cameron and Liebler. European J. Combin., 1999, vol. 20, no. 4, pp. 263–269. doi: 10.1006/eujc.1998.0265 .
5. Bruen A.A., Drudge K. The construction of Cameron-Liebler line classes in PG(3,q). Finite Fields Appl., 1999, vol. 5, no. 1, pp. 35–45. doi: 10.1006/ffta.1998.0239 .
6. De Beule J., Demeyer J., Metsch K., Rodgers M. A new family of tight sets in Q+(5,q). Des. Codes Cryptogr., 2016, vol. 78, no. 3, pp. 655–678. doi: 10.1007/s10623-014-0023-9 .
7. Feng T., Momihara K., Xiang Q. Cameron-Liebler line classes with parameter $x = (q^2-1)/2$. J. Combin. Theory Ser. A., 2015, vol. 133, pp. 307–338. doi: 10.1016/j.jcta.2015.02.004 .
8. Gavrilyuk A.L., Matkin I., Penttila T. Derivation of Cameron-Liebler line classes. Des. Codes Cryptogr., 2018, vol. 86, no. 1, pp. 231–236. doi: 10.1007/s10623-017-0338-4 .
9. Gavrilyuk A.L., Metsch K. A modular equality for Cameron-Liebler line classes. J. Combin. Theory Ser. A., 2014, vol. 127, pp. 224–242. doi: 10.1016/j.jcta.2014.06.004 .
10. Govaerts P., Penttila T. Cameron-Liebler line classes in PG(3,4). Bull. Belg. Math. Soc. Simon Stevin, 2006, vol. 12, no. 5, pp. 793–804.
11. Rodgers M. Cameron-Liebler line classes. Des. Codes Cryptogr., 2013, vol. 68, no. 1–3, pp. 33–37. doi: 10.1007/s10623-011-9581-2 .
12. Drudge K. Extremal sets in projective and polar spaces.: Thesis (Ph.D.)-The University of Western Ontario (Canada). 1998, 111 p.
13. Gavrilyuk A.L., Mogilnykh I.Yu. Cameron-Liebler line classes in PG(n,4). Des. Codes Cryptogr., 2014, vol. 73, no. 3, pp. 969–982. doi: 10.1007/s10623-013-9838-z .
14. Gavrilyuk A.L., Matkin I. Cameron–Liebler line classes in PG(3,5) [e-resource]. 19 p. Available at: https://arxiv.org/abs/1803.10442 .
