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Zbl 1028.14019
Freudenburg, Gene; Moser-Jauslin, Lucy
A nonlinearizable action of $S_3$ on $\bbfC^4$. (English)
[J] Ann. Inst. Fourier 52, No.1, 133-143 (2002). ISSN 0373-0956; ISSN 1777-5310/e

An action of a group $G$ on ${\Bbb C}^n$ by polynomial automorphisms is called linearizable if it is conjugate to a linear action under a polynomial automorphism. In this paper, the authors construct an action of the symmetric group $S_3$ on ${\Bbb C}^4$.\par {\it G. W. Schwarz} [C. R. Acad. Sci., Paris, Sér. I 309, 89-94 (1989; Zbl 0688.14040)] gave the first example of a nonlinearizable action of a reductive group on an affine space. He constructed a nonlineariable action of the orthogonal group $O(2)$ on ${\Bbb C}^4$ using equivariant vector bundles over an affine space. It is known that for $n\leq 2$ every action by a finite group action is linearizable. For $n=3$ it is still open whether non-linearizable finite group actions exist. For $n\geq 4$ there are examples of nonlinearizable finite group actions by {\it M. Masuda} and {\it T. Petrie} [J. Am. Math. Soc. 8, 687-714 (1995; Zbl 0862.14009)] ($D_{10}$ and larger groups) and Mederer [Thesis (Brandeis Univ. 1995)] ($D_5$ and $D_6$).\par The approach of equivariant vector bundles fails for Abelian groups. The smallest possible group for which this approach can work is $S_3$. In this paper the authors show that there does indeed exist a nonlinearizable action of $S_3$. Their example is a restriction of the example of Schwarz.
[Harm Derksen (Ann Arbor)]

MSC 2000:: *14L30 Group actions on varieties or schemes
13A50 Invariant theory
14R20 Group actions on affine varieties

Keywords: linearization problem; group actions; equivariant vector bundles

Citations: Zbl 0688.14040; Zbl 0862.14009

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