Class Groups of Multiplicative Invariants

AbstractLet G be a finite subgroup of GLd(Z). 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 (≅ Zd). We show that the class group of the ring of invariants of this action is isomorphic to Hom(G/N, k*) ⊕ H1(G/D, (Zd)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 Z