arXiv subjects

Hanspeter Kraft

Publications and source records attributed to Hanspeter Kraft.

21 records · Page 2Linked to original sources

arXiv2008

Compression of Finite Group Actions and Covariant Dimension, II

Let $G$ be a finite group and $ϕ: V\to W$ 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 \cite{KS07} we investigated covariant dimension and were able to determine it in many cases. Our techniques largely depended upon finding homogeneous faithful covariants. After publication of \cite{KS07}, the junior author of this article pointed out several gaps in our proofs. Fortunately, this inspired us to find better techniques, involving multihomogeneous covariants, which have enabled us to extend and complete the results, simplify the proofs and fill the gaps of \cite{KS07}.

math.AG↗

arXiv2006

A result of Hermite and equations of degree 5 and 6

A classical result from 1861 due to Hermite says that every separable equation of degree 5 can be transformed into an equation of the form x^5 + b x^3 + c x + d = 0. Later this was generalized to equations of degree 6 by Joubert. We show that both results can be understood as an explicit analysis of certain covariants of the symmetric groups S_5 and S_6. In case of degree 5, the classical invariant theory of binary forms of degree 5 comes into play whereas in degree 6 the existence of an outer automorphism of S_6 plays an essential role.

math.AC↗

arXiv2006

Compression of Finite Group Actions and Covariant Dimension

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 $n\geq 5$.

math.RT↗