Классы Камерона --- Либлера прямых в $\mathrm{PG}(n,5)$

Cameron‐Liebler line classes in PG(3, 5)

“…Further, the set-stabiliser of 12 (2) has the only orbit on lines that should have pattern T 6 , so, in the plane π 1 , we can consider an arbitrary configuration of 12 lines that is projectively equivalent to 12 (2) and contains the line ℓ ⋆ (as it is represented in Table 1). The setstabiliser of 18 has the only orbit on lines that should have pattern T 6 . In the plane π 2 , we consider all configurations of 18 lines that are projectively equivalent to 18 and contain the line ℓ ⋆ as it is represented in Table 1, and that are nonequivalent under the action of the point-wise stabiliser of ℓ ⋆ in PG (2,5).…”

“…The setstabiliser of 18 has the only orbit on lines that should have pattern T 6 . In the plane π 2 , we consider all configurations of 18 lines that are projectively equivalent to 18 and contain the line ℓ ⋆ as it is represented in Table 1, and that are nonequivalent under the action of the point-wise stabiliser of ℓ ⋆ in PG (2,5). This gives four candidates for a planar section of ′ in π 2 .…”

“…This theorem can be used to show that all Cameron-Liebler line classes in n PG( , 5), n > 3, are trivial, see [18]. We were informed by John Bamberg (a private communication) that he found all nontrivial Cameron-Liebler line classes in PG(3, 5) with a stabiliser divisible by an element of order 3, 5, 13, or 31.…”