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Pseudo-reflection group actions on local rings

Published online by Cambridge University Press:  22 January 2016

Luchezar L. Avramov
Luchezar L. Avramov*
Affiliation:
Institute of Mathematics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
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In a classical paper [C] Chevalley considered the invariants of a finite group H ⊂ GLk(S1) generated by pseudo-reflections, acting on the graded polynomial ring S = k[X1,…,Xn] over a field k of characteristic zero. He proved that S is free as a graded SH-module, hence SH is a graded polynomial ring (Theorem A), and that the natural representation of H in is equivalent to the regular representation (Theorem B). On the other hand, a theorem of Shephard and Todd shows that when SH is a polynomial ring, the (finite) group H is generated by pseudo-reflections. These results have been extended by Bourbaki [Bo2] to fields whose characteristic may be positive, but does not divide the order |H| of the group.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 88 , December 1982 , pp. 161 - 180
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1982

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