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A nonlinearizable action of $$S_3$$ on $$\mathbb{C}^4$$. (English) Zbl 1028.14019
An action of a group $$G$$ on $${\mathbb C}^n$$ 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 $${\mathbb C}^4$$.
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 $${\mathbb C}^4$$ 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 M. Masuda and 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 $$D_6$$).
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.

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 13A50 Actions of groups on commutative rings; invariant theory 14R20 Group actions on affine varieties
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