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Modular Vector Invariants of Cyclic Permutation Representations

  Published:1999-03-01
 Printed: Mar 1999
  • Larry Smith
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Abstract

Vector invariants of finite groups (see the introduction for an explanation of the terminology) have often been used to illustrate the difficulties of invariant theory in the modular case: see, \eg., \cite{Ber}, \cite{norway}, \cite{fossum}, \cite{MmeB}, \cite{poly} and \cite{survey}. It is therefore all the more surprising that the {\it unpleasant} properties of these invariants may be derived from two unexpected, and remarkable, {\it nice} properties: namely for vector permutation invariants of the cyclic group $\mathbb{Z}/p$ of prime order in characteristic $p$ the image of the transfer homomorphism $\Tr^{\mathbb{Z}/p} \colon \mathbb{F}[V] \lra \mathbb{F}[V]^{\mathbb{Z}/p}$ is a prime ideal, and the quotient algebra $\mathbb{F}[V]^{\mathbb{Z}/p}/ \Im (\Tr^{\mathbb{Z}/p})$ is a polynomial algebra on the top Chern classes of the action.
Keywords: polynomial invariants of finite groups polynomial invariants of finite groups
MSC Classifications: 13A50 show english descriptions Actions of groups on commutative rings; invariant theory [See also 14L24] 13A50 - Actions of groups on commutative rings; invariant theory [See also 14L24]
 

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