论文标题
有限通用线性组的交点定理
Intersection theorems for finite general linear groups
论文作者
论文摘要
通用线性$ \ permatorName {gl}（n，q）$的子集$ y $，如果$ \ operatatorName {rk}（x-y）（x-y）\ le n-t $ in y $ in y $ in y $或y $ x $ x $ x $ x $ x $ toppace $ \ mathbb {f} _q^n $ for All $ x，y \ in y $。我们表明，如果$ n $与$ t $相比，那么每一个这样的$ t $ iCTERTINCTing集的大小最多是$ \ mathbb {f} _q^n $的$ t $尺寸子空间的稳定器。在平等的情况下，$ y $的特征向量是这些稳定器coset的特征向量的线性组合。我们还为$ \ operatorname {gl}（n，q）的子集提供了类似的结果，该子集不一定在$ \ mathbb {f} _q^n $的$ t $二维子空间中与$ \ mathbb {f} _q^n $和交叉相互交流的子集$ \ \ \ \ \ \ \ \ operatateRatOrnArname {gl}（gl}（n，q）$相交。这些结果可能被视为极值集理论中经典的ERDőS-KO-RADO定理的变体，并且是对称组已知的相应结果的$ Q $ - Analogs。我们的方法基于特征值技术来估计图中最大的独立集的大小，并且至关重要的是涉及$ \ operatatorName {gl}（n，q）$的表示理论。
A subset $Y$ of the general linear group $\operatorname{GL}(n,q)$ is called $t$-intersecting if $\operatorname{rk}(x-y)\le n-t$ for all $x,y\in Y$, or equivalently $x$ and $y$ agree pointwise on a $t$-dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$. We show that, if $n$ is sufficiently large compared to $t$, the size of every such $t$-intersecting set is at most that of the stabiliser of a basis of a $t$-dimensional subspace of $\mathbb{F}_q^n$. In case of equality, the characteristic vector of $Y$ is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $\operatorname{GL}(n,q)$ that intersect not necessarily pointwise in $t$-dimensional subspaces of $\mathbb{F}_q^n$ and for cross-intersecting subsets of $\operatorname{GL}(n,q)$. These results may be viewed as variants of the classical Erdős-Ko-Rado Theorem in extremal set theory and are $q$-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $\operatorname{GL}(n,q)$.
