Speaker：Qing Xiang（South University of Science and Technology of China）

Time：2022-06-16, 15:00

Location：Tencent Meeting ID：375195443（No Pwd）

Abstract:

Cameron-Liebler line classes are sets of lines in PG(3,q) having many interesting combinatorial properties. These line classes were first introduced by Cameron and Liebler in their study of collineation groups of PG(3,q) having the same number of orbits on points and lines of PG(3,q). During the past decade, Cameron-Liebler line classes have received considerable attention from researchers in both finite geometry and algebraic combinatorics. In the original paper [1] by Cameron and Liebler, the authors gave several equivalent conditions for a set of lines of PG(3,q) to be a Cameron-Liebler line class; later Penttila gave a few more of such characterizations. We will use one of these characterizations as the definition of Cameron-Liebler line class. Let L be a set of lines of PG(3,q) with |L|=x(q^2+q+1), x a positive integer. We say that L is a Cameron-Liebler line class with parameter x if every spread of PG(3,q) contains x lines of L. It turned out that Cameron-Liebler line classes are closely related to certain subsets of points (tight sets) of the Klein quadric. We will talk about a recent construction in [2] of a new infinite family of Cameron-Liebler line classes with parameter x=(q+1)^2 /3 for q≡2(mod 3). When q is an odd power of 2, this family of Cameron-Liebler line classes represents the first infinite family of Cameron-Liebler line classes ever constructed in  PG(3,q), q even. This talk is based on joint work with Tao Feng, Koji Momihara, Morgan Rodgers and Hanlin Zou.

参考文献

[1] P. J. Cameron, R. A. Liebler, Tactical decompositions and orbits of projective groups, Linear Algebra Appl. 46 (1982), 91-102.

[2] T. Feng, K. Momihara, M. Rodgers, Q. Xiang, H. Zou, Cameron-Liebler line classes with parameter x=(q+1)^2/3, Advances in Math.  385 (2021), 107780.