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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A new degree bound for vector invariants
of symmetric groups


Author: P. Fleischmann
Journal: Trans. Amer. Math. Soc. 350 (1998), 1703-1712
MSC (1991): Primary 13A50
MathSciNet review: 1451600
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Abstract: Let $R$ be a commutative ring, $V$ a finitely generated free $R$-module and $G\le GL_R(V)$ a finite group acting naturally on the graded symmetric algebra $A=S(V)$. Let $\beta(V,G)$ denote the minimal number $m$, such that the ring $A^G$ of invariants can be generated by finitely many elements of degree at most $m$.

For $G=\Sigma _n$ and $V(n,k)$, the $k$-fold direct sum of the natural permutation module, one knows that $\beta(V(n,k),\Sigma _n) \le n$, provided that $n!$ is invertible in $ R$. This was used by E. Noether to prove $\beta(V,G) \le |G|$ if $|G|! \in R^*$.

In this paper we prove $\beta(V(n,k),\Sigma _n) \le max\{n,k(n-1)\}$ for arbitrary commutative rings $R$ and show equality for $n=p^s$ a prime power and $R = \mathbb Z$ or any ring with $n\cdot 1_R=0$. Our results imply

\begin{equation*}\beta(V,G)\le max\{|G|, \operatorname{rank}(V)(|G|-1)\}\end{equation*}

for any ring with $|G| \in R^*$.


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Additional Information

P. Fleischmann
Affiliation: Institute for Experimental Mathematics, University of Essen, Ellernstr. 29, 45326 Essen, Germany
Email: peter@exp-math.uni-essen.de

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02064-9
PII: S 0002-9947(98)02064-9
Received by editor(s): June 20, 1996
Article copyright: © Copyright 1998 American Mathematical Society