论文标题
Cameron-Liebler Line类
Cameron-Liebler line classes
论文作者
论文摘要
$ \ mathrm {pg}（3，q）$中Cameron -Liebler Line类的新示例用参数$ \ frac {1} {1} {2}（q^2 -1）$给出。这些示例是通过计算机搜索为$ q $的许多奇数值构建的，它是通过在空间上作用的环形胶体组组成的线条轨道结合的。虽然这些对象有许多等效的特征，但也许最重要的是，一组$ \ mathcal {l} $ in $ \ mathrm {pg}（pg}（3，q）$是cameron-liebler line类，具有参数$ x $的情况，并且只有当每个spread $ x $ and varrial $ x $ and sprail $ \ shotcal $ \ shatcal {s} $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x。这些对象与$ \ mathrm {pg}（3，q）$的对称战术分解的概括以及$ \ mathrm {pγl}（4，q）$的子组具有同样多的orbits On $ \ mathrm {pg}（pg}（3，q）$。此外，在某些情况下，我们构建的行类别与$ \ mathrm {ag}（2，q）$中的两次交流集有关。由于对于$ q $奇数的这些集很少有已知示例，因此在这个方向上的任何新结果都特别感兴趣。
New examples of Cameron-Liebler line classes in $\mathrm{PG}(3,q)$ are given with parameter $\frac{1}{2}(q^2 -1)$. These examples have been constructed for many odd values of $q$ using a computer search, by forming a union of line orbits from a cyclic collineation group acting on the space. While there are many equivalent characterizations of these objects, perhaps the most significant is that a set of lines $\mathcal{L}$ in $\mathrm{PG}(3,q)$ is a Cameron-Liebler line class with parameter $x$ if and only if every spread $\mathcal{S}$ of the space shares precisely $x$ lines with $\mathcal{L}$. These objects are related to generalizations of symmetric tactical decompositions of $\mathrm{PG}(3,q)$, as well as to subgroups of $\mathrm{PΓL}(4,q)$ having equally many orbits on points and lines of $\mathrm{PG}(3,q)$. Furthermore, in some cases the line classes we construct are related to two-intersection sets in $\mathrm{AG}(2,q)$. Since there are very few known examples of these sets for $q$ odd, any new results in this direction are of particular interest.
