On Multiplicative Invariants of Finite Reflection Groups

Abstract

This article focuses on two recent results on multiplicative invariants of finite reflection groups: Lorenz (2001) showed that such invariants are affine normal semigroup algebras, and Reichstein (2003) proved that the invariants have a finite SAGBI basis. Reichstein (2003) also showed that, conversely, if the multiplicative invariant algebra of a finite group G has a SAGBI basis, then G acts as a reflection group. There is no obvious connection between these two results. We will show that multiplicative invariants of finite reflection groups have a certain embedding property that implies both results simultaneously.

Key Words:

• Monomial order
• Multiplicative action
• Reflection group
• SAGBI basis
• Semigroup algebra

2000 Mathematics Subject Classification:

• 13A50
• 16W20
• 16S34
• 20H15

ACKNOWLEDGMENTS

A portion of this article is taken from my Ph.D. thesis. On this occasion I'd like to thank my supervisor Professor Martin Lorenz for all his support and advice to this day and in preparation of this manuscript. I would also like to thank the anonymous referee for the several valuable comments and corrections.

Notes

Communicated by J. Alev.