$\mathbb{F}_q^n$的最小循环覆盖子空间与Isbell猜想的下界
Smallest cyclically covering subspaces of $\mathbb{F}_q^n$, and lower bounds in Isbell's conjecture
摘要：
对于一个素数幂$q$和一个正整数$n$，如果$U$的循环移位的并等于$\mathbb{F}_q^n$，则称${\mathbb{F}_q^n}$的子空间$U$是{ \em循环覆盖}.研究了$\mathbb{F}_q^n$的循环覆盖子空间的最小可能维数的确定问题。(This是第一作者在1991年提出的问题的自然概括。我们证明了几个上限和下界，并为每个固定的$q$，我们完全回答了这个问题的无穷多个值的$n$（采取某些几何级数的形式）。我们的结果意味着下界的一个著名的猜想Isbell，和一个推广定理，补充下界由于Spiga。我们还考虑了群的一般表示的类似问题。我们使用的参数从组合，表示论和有限域理论。
英文摘要：
For a prime power $q$ and a positive integer $n$, we say a subspace $U$ of ${\mathbb{F}_q^n}$ is {\em cyclically covering} if the union of the cyclic shifts of $U$ is equal to $\mathbb{F}_q^n$. We investigate the problem of determining the minimum possible dimension of a cyclically covering subspace of $\mathbb{F}_q^n$. (This is a natural generalisation of a problem posed in 1991 by the first author.) We prove several upper and lower bounds, and for each fixed $q$, we answer the question completely for infinitely many values of $n$ (which take the form of certain geometric series). Our results imply lower bounds for a well-known conjecture of Isbell, and a generalisation theoreof, supplementing lower bounds due to Spiga. We also consider the analogous problem for general representations of groups. We use arguments from combinatorics, representation theory and finite field theory.
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