The infiniteness of the SAGBI bases for certain invariant rings.
type: Item
instance of: scholarly article
title: The infiniteness of the SAGBI bases for certain invariant rings.
zbMATH Open document ID: 1045.13012
author: Shigeru Kuroda
published in: Osaka Journal of Mathematics
publication date: 2002-12-10T00:00:00Z
review text: The concept of initial ideals for ideals of a polynomial ring in Gröbner bases theory is generalized in a natural way for subalgebras of a polynomial ring, and they are called initial algebras. A set of generators of a subalgebra is called a SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis if their initial monomials generate the initial algebra. The main difference between the initial ideal and the initial algebra is that the former always is finitely generated while the latter does not. Hence it is an important problem to find a criterion for the finite generation of initial algebras. \textit{M. Göbel} [J. Symb. Comput. 26, 261--272 (1998; Zbl 0916.13011)], studied this problem for the subalgebra of invariants of a permutation group \(G\). He proved that, with respect to the lexicographic order, the initial algebra of \(K[V]^G\) is finitely generated if and only if \(G\) is a direct product of symmetric groups. In the present paper, the same result is proved for any multiplicative order, i.e., a monomial order which does not require the minimality of the unit \(1\), and also for the action on the Laurent polynomial ring. A principal role in the proof plays the topological structure on the set of multiplicative orders. Using this structure and the Baire theorem, the author proves that there exist uncountable cardinality of distinct initial algebras for each invariant ring, when \(G\) is not a direct product of symmetric groups. (Note that in the case of initial ideals, there exist only a finite cardinality of distinct initial ideals for an ideal under a certain condition, although there exist infinitely many orders.) If \(G\) is a product of symmetric groups, the number of distinct initial algebras of the subalgebra of invariants is finite and is equal to the order of \(G\).
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 14R20
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 1836041
zbMATH Keywords: subalgebra analogue to Gröbner bases for ideals
zbMATH Keywords: SAGBI
zbMATH Keywords: Laurent polynomial
zbMATH Keywords: polynomial invariants
zbMATH Keywords: finite permutation groups
zbMATH Keywords: initial terms
zbMATH Keywords: multiplicative orders
zbMATH Keywords: finite generation of initial algebras
zbMATH Keywords: subalgebra of invariants of a permutation group
reviewed by: Ivan V. Arzhantsev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1836041