Homomorphisms, Localizations and a new Algorithm to construct Invariant Rings of Finite Groups

Fleischmann, Peter and Kemper, Gregor and Woodcock, Chris F. (2007) Homomorphisms, Localizations and a new Algorithm to construct Invariant Rings of Finite Groups. Journal of Algebra, 309 (2). pp. 497-517. ISSN 0021-8693. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

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Official URL
http://dx.doi.org/10.1016/j.jalgebra.2005.06.038

Abstract

Let G be a finite group acting on a polynomial ring A over the field K and let AG denote the corresponding ring of invariants. Let B be the subalgebra of AG generated by all homogeneous elements of degree less than or equal to the group order |G|. Then in general B is not equal to AG if the characteristic of K divides |G|. However we prove that the field of fractions Quot(B) coincides with the field of invariants Quot(AG)=Quot(A)G. We also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for AG. We prove that there is always a nonzero transfer cAG of degree <|G|, such that the localization (AG)c can be generated by fractions of homogeneous invariants of degrees less than 2|G|−1. If with finite-dimensional -module V, then c can be chosen in degree one and 2|G|−1 can be replaced by |G|. Let denote the image of the classical Noether-homomorphism (see the definition in the paper). We prove that contains the transfer ideal and thus can be used to calculate generators for AG by standard elimination techniques using Gröbner-bases. This provides a new construction algorithm for AG.

Item Type: Article
Uncontrolled keywords: Modular invariant theory; Computational algebra; Localization
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science
Depositing User: Judith Broom
Date Deposited: 19 Dec 2007 18:27
Last Modified: 19 May 2014 13:26
Resource URI: https://kar.kent.ac.uk/id/eprint/732 (The current URI for this page, for reference purposes)
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