有限群作用的压缩与协变维数
Compression of Finite Group Actions and Covariant Dimension
摘要：
设$G$是有限群，$ϕ\colon V\to W$是有限维$G$ -模的等变态射.如果$G$忠实地作用于$ϕ(V)$，我们说$ϕ$是忠实的。$G$的协变维数是在所有忠实的$ϕ$上取的$\bar{ϕ(V)}$的维数的最小值。在本文中，我们研究协变维数，并能够确定它的阿贝尔群，并获得估计的对称和交替的群体。我们还分类组的协变维数小于或等于2。一个副产品，我们的调查是存在一个纯粹的先验领域的定义度$n-3$的一般域扩张度$n\geq 5$。
英文摘要：
Let $G$ be a finite group and $ϕ\colon V\to W$ an equivariant morphism of finite dimensional $G$-modules. We say that $ϕ$ is faithful if $G$ acts faithfully on $ϕ(V)$. The covariant dimension of $G$ is the minimum of the dimension of $\bar{ϕ(V)}$ taken over all faithful $ϕ$. In this paper we investigate covariant dimension and are able to determine it for abelian groups and to obtain estimates for the symmetric and alternating groups. We also classify groups of covariant dimension less than or equal to 2. A byproduct of our investigations is the existence of a purely transcendental field of definition of degree $n-3$ for a generic field extension of degree $n\geq 5$.
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