Jurkiewicz, Jerzy On the linearization of actions of linearly reductive groups. (English) Zbl 0698.14053 Comment. Math. Helv. 64, No. 3, 508-513 (1989). Let G be a linearly reductive group acting on an affine space \(A^ n\). Suppose G has a fixed point and acts by polynomial transformations of degree two. If the ground field k has characteristic two, assume moreover that G is abelian. Then it is proved that the action is linearizable. An explicit formula, using the Reynolds operator, is given for an automorphism of \(A^ n\), “linearizing” the action. Remark: 1. If char k\(=0\), then the claim has been recently proved under weaker assumption [see the author’s paper “On some reductive group actions on affine space” in Group actions and invariant theory, Proc. Conf., Montreal/Can. 1988, Conf. Proc. 10, 67-72 (1989). \(-\quad 2.\quad In\) general the action of reductive groups on affine space is not linearizable, see G. W. Schwarz, C. R. Acad. Sci. Paris, Sér. I Math. 309, No.2, 89-94 (1989; Zbl 0688.14040)]. Reviewer: J.Jurkiewicz Cited in 1 ReviewCited in 3 Documents MSC: 14L30 Group actions on varieties or schemes (quotients) 20G40 Linear algebraic groups over finite fields Keywords:linearization of actions of linearly reductive groups; characteristic two; Reynolds operator Citations:Zbl 0688.14040 PDFBibTeX XMLCite \textit{J. Jurkiewicz}, Comment. Math. Helv. 64, No. 3, 508--513 (1989; Zbl 0698.14053) Full Text: DOI EuDML