论文标题
$ \ text {ag}（n，q）$的cameron-liebler $ k $ -set
Cameron-Liebler $k$-sets in $\text{AG}(n,q)$
论文作者
论文摘要
我们研究仿射几何形状中的Cameron-Liebler $ k $ -set，因此$ \ text {ag}（ag}（n，q）$的$ k $ spaces集合。这概括了对投影几何$ \ text {pg}（n，q）$的Cameron-Liebler $ K $ -Set的研究。请注意，在代数组合学中，Cameron-liebler $ k $ - $ \ text {ag}（n，q）$的$对应于$ k $ spaces in $ \ text {ag ag}（ag}（n，q）$的某些公平两分，同时，在布尔函数的分析中，$ 1 $ $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ 1 $ $ 1 $ 1 $ 1 $ $ 1 Q）$。我们通过与$ k $ spreads的交集属性定义$ \ text {ag}（n，q）$中的cameron-liebler $ k $ -sets（n，q）$，并显示了几个定义的等效性。特别是，我们研究了$ \ text {ag}（n，q）$和$ \ text {pg}（n，q）$中的cameron-liebler $ k $ -set之间的关系。作为副产品，我们计算仿期线的关联方案的特征表。此外，我们描述了Cameron-Liebler $ k $ -set的最小示例。本文重点介绍$ n> 3 $的$ \ text {ag}（n，q）$，而$ \ text {ag}（3，q）$中的Cameron-Liebler Line类的情况已经单独处理。
We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\text{AG}(n, q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\text{PG}(n, q)$. Note that in algebraic combinatorics, Cameron-Liebler $k$-sets of $\text{AG}(n, q)$ correspond to certain equitable bipartitions of the Association scheme of $k$-spaces in $\text{AG}(n, q)$, while in the analysis of Boolean functions, they correspond to Boolean degree $1$ functions of $\text{AG}(n, q)$. We define Cameron-Liebler $k$-sets in $\text{AG}(n, q)$ by intersection properties with $k$-spreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler $k$-sets in $\text{AG}(n, q)$ and $\text{PG}(n, q)$. As a by-product, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler $k$-sets. This paper focuses on $\text{AG}(n, q)$ for $n > 3$, while the case for Cameron-Liebler line classes in $\text{AG}(3, q)$ was already treated separately.
