SAGBI bases in rings of multiplicative invariants

Abstract

Let k be a field and G be a finite subgroup of ${\mathrm{GL}}_n(\mathbb{Z})$. 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.