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arXiv 1009.1997math.CO
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对偶极空间中公寓的度量刻画

Metric characterization of apartments in dual polar spaces

Mark Pankov

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摘要：

设$Π$是秩为$n$的极空间，${\mathcal G}_{k}(Π)$，$k\in \{0,\dots,n-1\}$是由$Π$的$k$维奇异子空间构成的极格拉斯曼空间.相应的格拉斯曼图将由$Γ_{k}(Π)$表示。本文研究了由$Π$的极大奇异子空间构成的极Grassmannian ${\mathcal G}_{n-1}(Π)$，证明了$n$维超立方体图$H_{n}$在$Γ_{n-1}(Π)$中的等距嵌入的象是${\mathcal G}_{n-1}(Π)$的一个公寓.这是从关于$H_{m}$、$m\le n$在$Γ_{n-1}(Π)$中的等距嵌入的更一般的结果（定理2）得出的。作为应用，我们对$Γ_{n-1}(Π)$在$Γ_{n'-1}(Π')$中的等距嵌入进行了分类，其中$Π'$是秩为$n'\ge n$的极空间（定理3）.

英文摘要：

Let $Π$ be a polar space of rank $n$ and let ${\mathcal G}_{k}(Π)$, $k\in \{0,\dots,n-1\}$ be the polar Grassmannian formed by $k$-dimensional singular subspaces of $Π$. The corresponding Grassmann graph will be denoted by $Γ_{k}(Π)$. We consider the polar Grassmannian ${\mathcal G}_{n-1}(Π)$ formed by maximal singular subspaces of $Π$ and show that the image of every isometric embedding of the $n$-dimensional hypercube graph $H_{n}$ in $Γ_{n-1}(Π)$ is an apartment of ${\mathcal G}_{n-1}(Π)$. This follows from a more general result (Theorem 2) concerning isometric embeddings of $H_{m}$, $m\le n$ in $Γ_{n-1}(Π)$. As an application, we classify all isometric embeddings of $Γ_{n-1}(Π)$ in $Γ_{n'-1}(Π')$, where $Π'$ is a polar space of rank $n'\ge n$ (Theorem 3).

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