SAGBI bases in rings of multiplicative invariants
type: Item
instance of: scholarly article
title: SAGBI bases in rings of multiplicative invariants
zbMATH Open document ID: 1043.13008
author: Zinovy Reichstein
published in: Commentarii Mathematici Helvetici
publication date: 2003-10-12T00:00:00Z
review text: The well-known concept of a Gröbner basis for a polynomial ideal over a field helps to decide a great number of important questions about these ideals. Every polynomial ideal has a finite Gröbner basis which can be effectively determined. If we consider subrings of the polynomial ring, then we can define a concept which is close but not identical to a Gröbner basis: a so-called SAGBI (Subalgebra Analog to Gröbner Basis for Ideals) basis. One of the important questions related to SAGBI bases is which rings have a finite SAGBI basis. Z. Reichstein considers a slightly more general situation, namely the polynomial ring over a field in finitely many variables and their inverses, i.e. Laurent polynomials. He proves that if \(R\) is the ring of multiplicative invariants for a finite subgroup \(G\) of the general linear group over the integers, then the following are equivalent: (a) the semigroup of initials of \(R\) is finitely generated, (b) \(R\) has a finite SAGBI basis, and (c) \(G\) is a reflection group.
reviewed by: Franz Winkler
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 1990091
zbMATH Keywords: finite SAGBI basis
zbMATH Keywords: Laurent polynomials
zbMATH Keywords: invariants for general linear group
Wikidata QID: Q117379615
MaRDI profile type: Publication
full work available at URL: s000140300008
OpenAlex ID: W2050554783
DOI: 10.1007/S000140300008
description: scientific article; zbMATH DE number 1990091