# Commentarii Mathematici Helvetici

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**Volume 78, Issue 1, 2003, pp. 185–202**

**DOI: 10.1007/s000140300008**

Published online: 2003-03-31

SAGBI bases in rings of multiplicative invariants

Zinovy Reichstein^{[1]}(1) University of British Columbia, Vancouver, Canada

Let k be a field and G be a finite subgroup of $\GL_n(\mathbb Z)$. We show that the ring of multiplicative invariants $k[x_1^{\pm 1}, \dots, x_n^{\pm 1}]^G$ has a finite SAGBI basis if and only if G is generated by reflections.

*Keywords: *SAGBI basis, term order, initial term, subduction algorithm, Gröbner basis, multiplicative groups action, algebra of invariants, integral representation, reflection group, permutation group, convex cone, polyhedral cone, finitely generated semigroup

Reichstein Zinovy: SAGBI bases in rings of multiplicative invariants. *Comment. Math. Helv.* 78 (2003), 185-202. doi: 10.1007/s000140300008