论文标题
经典空间中非分类互补子空间的比例
The proportion of non-degenerate complementary subspaces in classical spaces
论文作者
论文摘要
给定正整数$ e_1，e_2 $，令$ x_i $表示固定有限矢量空间的$ e_i $ - 维度子空间的集合，令$ y_i $为$ x_i $的非空子集，让$α_i= | y_i |/| x_i | $。我们给出一个正下限，仅取决于$α_1，α_2，e_1，e_2，q $，分别对$（s_1，s_2）\在y_1 \ times y_2 $中的比例（s_1，s_2）\。作为一种应用，我们在有限的经典空间中绑定了互补维度对互补维度对的比例。这个问题是由识别古典群体的算法激发的。通过使用代数图理论中的技术，我们能够在第2阶领域处理正交组，该案例使Niemeyer，Praeger和第一作者都无法使用。
Given positive integers $e_1,e_2$, let $X_i$ denote the set of $e_i$-dimensional subspaces of a fixed finite vector space $V=(\mathbb{F}_q)^{e_1+e_2}$. Let $Y_i$ be a non-empty subset of $X_i$ and let $α_i=|Y_i|/|X_i|$. We give a positive lower bound, depending only on $α_1,α_2,e_1,e_2,q$, for the proportion of pairs $(S_1,S_2)\in Y_1\times Y_2$ which intersect trivially. As an application, we bound the proportion of pairs of non-degenerate subspaces of complementary dimensions in a finite classical space that intersect trivially. This problem is motivated by an algorithm for recognizing classical groups. By using techniques from algebraic graph theory, we are able to handle orthogonal groups over the field of order 2, a case which had eluded Niemeyer, Praeger, and the first author.
