Compression of Finite Group Actions and Covariant Dimension

Details Compression of Finite Group Actions and Covariant Dimension Compression of Finite Group Actions and Covariant Dimension

2006-09-09

arxiv.org/abs/math/0609253v1

Mathematics

Representation Theory

22 pages

Scientific paper

Let $G$ be a finite group and $\phi\colon V\to W$ an equivariant morphism of finite dimensional $G$-modules. We say that $\phi$ is faithful if $G$ acts faithfully on $\phi(V)$. The covariant dimension of $G$ is the minimum of the dimension of $\bar{\phi(V)}$ taken over all faithful $\phi$. 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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