Rings of invariants of finite groups which are hypersurfaces. II
type: Item
instance of: scholarly article
title: Rings of invariants of finite groups which are hypersurfaces. II
zbMATH Open document ID: 0626.14010
DOI: 10.1016/0001-8708(87)90018-1
author: Haruhisa Nakajima
published in: Advances in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: [For part I see J. Algebra 80, 279-294 (1983; Zbl 0524.14013).] Let V be a finite dimensional complex vector space. An element f of \(GL(V)\) is a ``pseudo-reflection'' if f-id has rank one. Let H be a ``reflection subgroup of \(GL(V)\)'', i.e. a subgroup of GL(V) which is generated by pseudo-reflections. Then the ``order'' of a pseudo- reflection \(f\in H\) is the cardinality of \(\{g\in H| \quad Fix(g)\supseteq Fix(f)\}\) (where \(Fix(f):=\{v\in V| \quad f(v)=v\}.\) Main result: Let G be a finite subgroup of \(SL(V)\). Suppose that \(\dim_{{\mathbb{C}}}(V)>10\) and \(^ GV=\{0\}\). Then the quotient \(V/G\) is a hypersurface iff \(G=\tilde G\cap SL(V)\) for some finite reflection subgroup \(\tilde G\) of \(GL(V)\) in which all orders of pseudo-reflections are equal to the index \([\tilde G:G]\).
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 4017064
zbMATH Keywords: rings of invariants
zbMATH Keywords: finite linear groups
zbMATH Keywords: quotient varieties
zbMATH Keywords: pseudo- reflection
reviewed by: F. Pauer
MaRDI profile type: Publication
full work available at URL: 0001-8708(87)90018-1
OpenAlex ID: W2027351342
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description: scientific article; zbMATH DE number 4017064