SEMINAR: Groups and Combinatorics Seminar: Cameron-Liebler line classes and two-intersection sets

Time and place: 15:00 Friday 24 July in Austin LT.

Speaker: Qing Xiang (University of Delaware)

Title: Cameron-Liebler line classes and two-intersection sets

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). In the last few years, Cameron-Liebler line classes have received considerable attention from researchers in both finite geometry and algebraic combinatorics. In [1], the authors gave several equivalent conditions for a set of lines of PG(3,q) to be a Cameron-Liebler line class; 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 nonnegative 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 projective two-weight codes (equivalently, certain two-intersection sets in PG(5,q)).

We will talk about a recent construction of a new infinite family of Cameron-Liebler line classes with parameter x=(q^2-1)/2 for q = 5 or 9 (mod 12). This family of Cameron-Liebler line classes generalizes the examples found by Rodgers in through a computer search, and represents the second infinite family of Cameron-Liebler line classes. Furthermore, in the case where q is an even power of 3, we construct the first infinite family of affine two-intersection sets.

We should remark that De Beule, Demeyer, Metsch and Rodgers also independently obtained the same result as ours on Cameron-Liebler line classes with parameter x=(q^2-1)/2 at almost the same time.

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