Class Groups of Multiplicative Invariants Original Research Article

Let G be a finite subgroup of GLd(image). Then G acts on the Laurent polynomial ring k[X±11,..., X±1d] over the field k via the natural G-action on the multiplicative group generated by the variables X1,...,Xd (congruent with imaged). We show that the class group of the ring of invariants of this action is isomorphic to Hom(G/N, k*) circled plus H1(G/D, (imaged)D), where N denotes the subgroup of G that is generated by all reflections in G and D the subgroup generated by the reflections that are diagonalizable over image.