Speaker: Morgan Rodgers
Title: Cameron–Liebler line classes admitting large cyclic groups
Date/Time: 6 October 2021 / 13:40 - 14:30 PM
Zoom: Meeting ID: https://sabanciuniv.zoom.us/j/94368587993?pwd=WnpVZ3U1MTBBUGFHSTRRT0Flcmlhdz09
Passcode: 658402
Abstract: Cameron–Liebler line classes are special sets of lines in PG(3,q) with nice intersection properties; they were originally described in [1] in relation to line orbits of projective groups. We can think of them as extremal sets of lines having as many intersections as possible, or they can alternately be characterized as sets sharing a constant number of lines with every spread of the space. Such a set must contain precisely x(q2+ q + 1) lines for some integer x, called the parameter of the line class. There has been a lot of recent interest in Cameron–Liebler line classes, though there are only a small number of known families of examples. We will discuss some of the history of these objects, and then look at some recent examples constructed as unions of line orbits from a cyclic group of size q2 +q + 1. Using an algebraic model of projective space, we are able to study these objects and prove the existence of some infinite families using a connection with partial difference sets. This talk describes joint work appearing in [2] and [3].
References
[1] P.J. Cameron and R.A. Liebler. Tactical decompositions and orbits of projective groups. Linear Algebra and its Applications 46, 1982.
[2] J. De Beule, J. Demeyer, K. Metsch, and M. Rodgers. A new family of tight sets in Q+(5, q), Designs, Codes and Cryptography 78 (3), 2014.
[3] T. Feng, K. Momihara, M. Rodgers, Q. Xiang, and H. Zou. Cameron–Liebler line classes with parameter x =(q+1)2 /3 Advances In Mathematics 385, 2021.