关于紧密不可约仿射展布的一个注记

中文摘要 AI 辅助

设\({\mathbb F}_q^n\)表示\({\mathbb F}_q\)上维度为\(n\)的向量空间，AG\((n,q)\)表示相应的仿射空间。AG\((n,q)\)的仿射向量空间划分是一个仿射子空间的集合\({\mathcal P}\)，它划分了AG\((n,q)\)的点。若\({\mathcal P}\)中所有子空间具有相同维度\(d\)，则\({\mathcal P}\)称为仿射\(d -\)展布。若对于\({\mathcal P}\)中任意一对\(C,C'\)，\(C = v+S\)，\(C' = v'+S'\)（其中\(v,v'\in{\rm AG}(n,q)\)且\(S\neq S'\)是\({\mathbb F}_q^n\)的线性子空间），有\(S\cap S'=\{{\bf 0}\}\)，则仿射划分\({\mathcal P}\)是完全紧密的。若不存在子集\({\mathcal P}'\subset {\mathcal P}\)使得\(1<|{\mathcal P}'|<|{\mathcal P}|\)且\({\mathcal P}'\)中所有子空间的并是AG\((n,q)\)的子空间，则仿射划分\({\mathcal P}\)是不可约的。对于所有\(d \geq 1\)且\(n>2d\)，构造了AG\((n,q)\)的一个完全紧密不可约仿射\(d -\)展布，这也解决了Bamberg等人关于紧密不可约仿射\(d -\)展布存在性的一个近期猜想。

英文摘要

Let ${\mathbb F}_q^n$ denote the vector space of dimension $n$ over ${\mathbb F}_q$ and AG$(n,q)$ denote the corresponding affine space. An $\textit{affine vector space partition}$ of AG$(n,q)$ is a collection ${\mathcal P}$ of affine subspaces that partition the points of AG$(n,q)$. If all subspaces in ${\mathcal P}$ have the same dimension $d$, then ${\mathcal P}$ is called an $\textit{affine $d$-spread}$. We say that an affine partition ${\mathcal P}$ is $\textit{completely tight}$ if for any pair $C,C'\in{\mathcal P}$ with $C=v+S$, $C'=v'+S'$, where $v,v'\in{\rm AG}(n,q)$ and $S\neq S'$ are linear subspaces of ${\mathbb F}_q^n$, we have $ S\cap S'=\{{\bf 0}\}$. An affine partition ${\mathcal P}$ is said to be $\textit{irreducible}$ if there is no subset ${\mathcal P}'\subset {\mathcal P}$ such that $1<|{\mathcal P}'|<|{\mathcal P}|$ and the union of all subspaces in ${\mathcal P}'$ is a subspace of AG$(n,q)$. For all $d \geq 1$ and $n>2d$, we construct a completely tight irreducible affine $d$-spread of AG$(n,q)$. This also settles a recent conjecture of Bamberg et al. on the existence of tight irreducible affine $d$-spreads.

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