This article focuses on two recent results on multiplicative invariants of finite reflection groups: Lorenz (2001) showed that such invariants are aff. ne 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.
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