Computation of invariants for reductive groups
type: Item
instance of: scholarly article
title: Computation of invariants for reductive groups
zbMATH Open document ID: 0927.13007
published in: Advances in Mathematics
publication date: 1999-11-18T00:00:00Z
review text: Let \(G\) be a linear algebraic group defined over the field \(k\), \(\rho: G\to GL(V)\) a representation of \(G\) on a \(k\)-vector space of finite dimension \(n\) and \(A:=k[X_1, \dots, X_n]\) the coordinate ring of \(V\) with natural linear action of \(G\). Due to Hilbert, the invariant ring \(A^G: =\{a\in A\mid {}^ga=a\}\) is finitely generated if \(G\) is linearly reductive, i.e. if every finite dimensional \(kG\)-module is completely reducible. In the present paper the author presents an algorithm for computing generators of the invariant ring for arbitrary linearly reductive groups. His algorithm is based on Hilbert's original proof of 1890, which was considered non-constructive at that time. Under the hypotheses of certain conjectures in commutative algebra, the presented algorithm also gives improved upper bounds for the degrees of generators: In the case of \(k\) being of characteristic 0 and \(G\) linearly reductive of positive dimension, the degree bound significantly improves Popov's degree bound, but it depends on a conjecture of Eisenbud-Goto about degree bounds for syzygies of homogeneous prime ideals in \(A\). In the case of a finite group of order \(| G|\) not divisible by char \(k\), the author proves that Noether's degree bound holds, i.e. \(A^G\) can be generated by invariants of degree \(\leq| G|\), provided a so-called `subspaces conjecture' is true. Notice that Noether's bound is well known to be true for finite groups and \(\text{char }k=0\) of larger than \(| G|\). It is also known to be false for \(\text{char } k\mid| G|\). Recently the reviewer has proved with different methods, that Noether's bound holds whenever \(| G|\) is invertible in \(k\) [cf. \textit{P. Fleischmann}, Trans. Am. Math. Soc. 350, No. 4, 1703-1712 (1998; Zbl 0891.13002)].
reviewed by: Peter Fleischmann
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 13-04
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 1271738
zbMATH Keywords: reduction groups
zbMATH Keywords: polynomial invariants
zbMATH Keywords: algorithm
zbMATH Keywords: computing generators of the invariant ring
author: Harm Derksen
MaRDI profile type: Publication
OpenAlex ID: W2039361498
DOI: 10.1006/AIMA.1998.1787
description: scientific article; zbMATH DE number 1271738