The Construction of Cameron–Liebler Line Classes inPG(3,q)

Abstract

This paper is concerned with fundamental questions lying at the boundary of combinatorics, geometry, and group theory. For example, letAbe the point–line incidence matrix of Σ=PG(3,q). ThenAis a (0, 1)-matrix, with the columns (rows) corresponding to points (lines) of Σ. Working over Q, say, the following question is natural: “Which linear combinations of the columns are themselves (0, 1)-vectors?” There are some obvious examples. In fact, in 1982, Cameron and Liebler conjectured that only the obvious examples exist. Here we settle this 16-year-old conjecture in the negative.This question turns out to be intimately related to geometrical properties of certain line sets in Σ, and to studies of the orbit structure of the collineation groupPΓL(n+1,q) of Σ.