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

✍ Scribed by A.A. Bruen; Keldon Drudge

• Book ID: 102572835
• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 121 KB
• Volume: 5
• Category: Article
• ISSN: 1071-5797
• DOI: 10.1006/ffta.1998.0239

✦ Synopsis

This paper is concerned with fundamental questions lying at the boundary of combinatorics, geometry, and group theory. For example, let A be the point-line incidence matrix of "PG(3, q). Then A is a (0, 1)-matrix, with the columns (rows) corresponding to points (lines) of . Working over 0, 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 , and to studies of the orbit structure of the collineation group P ¸(n#1, q) of .