Z. Reichstein, SAGBI bases in rings of multiplicative invariants, Commentarii Math Helvetici 78 (2003), 185 -- 202.

Abstract: The subject matter of this paper falls somewhere between computational algebra, invariant theory, and the combinatorics of polyhedral cones in R^n. The main result is as follows. Let k be a field and G be a finite subgroup of GL_n(Z). We show that the ring of multiplicative invariants k[x_1, 1/x_1, \dots, x_n, 1/x_n]^G has a finite SAGBI basis if and only if G is generated by reflections.

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