SAGBI bases in rings of multiplicative invariants
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.
scientific article; zbMATH DE number 1990091