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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A new degree bound for vector invariants of symmetric groups
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by P. Fleischmann PDF
Trans. Amer. Math. Soc. 350 (1998), 1703-1712 Request permission

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
  • Received by editor(s): June 20, 1996
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 1703-1712
  • MSC (1991): Primary 13A50
  • DOI: https://doi.org/10.1090/S0002-9947-98-02064-9
  • MathSciNet review: 1451600