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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.

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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