# 組合せ論セミナー

## 第46回 2013年10月8日 15:30〜17:00

Alexander Gavrilyuk （東北大学）「Cameron-Liebler line classes in PG(n,q) 」

A Cameron-Liebler line class L with the parameter x, where x\leq (q^2+1)/2, is a set of lines of the projective geometry PG(3,q) such that each line of L meets exactly x(q+1)+q^2-1 lines of L and each line that is not from L meets exactly x(q+1) lines of L (this is only one of several equivalent definitions of Cameron-Liebler line classes).

These classes appeared in connection with an attempt by Cameron and Liebler (1982) to classify collineation groups of PG(n,q), n\geq 3, that have equally many orbits on lines and on points.

Cameron and Liebler conjectured that, apart from trivial examples with x in {0,1,2}, there are no Cameron-Liebler line classes. The conjecture was disproved by Drudge (1999) (at the moment there are several counterexamples known). On the other hand, Metsch (2010) showed that if x>2 then x>q (this is the best known lower bound for parameter x).

In this work, we derive a new existence condition for Cameron-Liebler line classes in PG(3,q). As an application, we obtain the classification of those in PG(3,4) (I will also concern a generalization of those in PG(n,q), n>3). This part is based on joint work with Ivan Mogilnykh.

Further, we show that the new existence condition eliminates about a half of the possible values for parameter x. Using this condition, we also construct a new counterexample to the Cameron-Liebler conjecture, namely, a line class with parameter x=10 in PG(3,5). This part is based on joint work with Klaus Metsch.