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 L of lines such that any line ℓ intersects x(q+1)+χL(ℓ)(qn−1+⋯+q2−1) lines from L, where χL is the characteristic function of the set 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.
\Bibitem{Mat18}
\by I.~Matkin
\paper Cameron-Liebler line classes in PG(n, 5)
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2018
\vol 24
\issue 2
\pages 158--172
\mathnet{http://mi.mathnet.ru/timm1531}
\crossref{https://doi.org/10.21538/0134-4889-2018-24-2-158-172}
\elib{https://elibrary.ru/item.asp?id=35060686}
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This publication is cited in the following 5 articles:
Ferdinand Ihringer, “The classification of Boolean degree 1 functions in high-dimensional finite vector spaces”, Proc. Amer. Math. Soc., 2024
Jan De Beule, Jonathan Mannaert, Leo Storme, “Cameron–Liebler k-sets in subspaces and non-existence conditions”, Des. Codes Cryptogr., 90:3 (2022), 633
E. A. Bespalov, D. S. Krotov, A. A. Matiushev, A. A. Taranenko, K. V. Vorob'ev, “Perfect 2-colorings of Hamming graphs”, J. Comb Des., 29:6 (2021), 367–396
Yu. Filmus, F. Ihringer, “Boolean degree 1 functions on some classical association schemes”, J. Comb. Theory Ser. A, 162 (2019), 241–270
Gavrilyuk A.L., Matkin I., “Cameron-Liebler Line Classes in Pg(3,5)”, J. Comb Des., 26:12 (2018), 563–580