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TitleHomomorphisms, localizations and a new algorithm to construct invariant rings of finite groups
Author(s) Peter Fleischmann, Gregor Kemper, Chris Woodcock
TypeArticle in Journal
AbstractLet G be a finite group acting on a polynomial ring A over the field K and let A G denote the corresponding ring of invariants. Let B be the subalgebra of A G generated by all homogeneous elements of degree less than or equal to the group order | G | . Then in general B is not equal to A G if the characteristic of K divides | G | . However we prove that the field of fractions Quot ( B ) coincides with the field of invariants Quot ( A G ) = Quot ( A ) G . We also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for A G . We prove that there is always a nonzero transfer c &#8712; A G of degree < | G | , such that the localization ( A G ) c can be generated by fractions of homogeneous invariants of degrees less than 2 &#8901; | G | &#8722; 1 . If A = Sym ( V &#8853; F G ) with finite-dimensional F G -module V, then c can be chosen in degree one and 2 &#8901; | G | &#8722; 1 can be replaced by | G | . Let N denote the image of the classical Noether-homomorphism (see the definition in the paper). We prove that N contains the transfer ideal and thus can be used to calculate generators for A G by standard elimination techniques using Gröbner-bases. This provides a new construction algorithm for A G .
KeywordsModular invariant theory, Computational algebra, Localization
ISSN0021-8693
URL http://www.sciencedirect.com/science/article/pii/S0021869305006435
LanguageEnglish
JournalJournal of Algebra
Volume309
Number2
Pages497 - 517
Year2007
NoteComputational Algebra
Edition0
Translation No
Refereed No
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