Reflection groups and multiplicative invariants (Q1082454)

Let M be a lattice (i.e. a finitely generated torsion free abelian group), C[M] its group algebra, \(G\subset GL(M)\) a finite reflection group, \(C[M]^ G\) the fixed ring under G, and C(M) the rational function field of fractions. The author shows, \(C(M)^ G\) is a rational function field; \(C[M]^ G\) is a polynomial ring over \(C[M^ G]\) if and only if for some choice of root system, \(M/M^ G\) can be realized as its weight lattice and the group induced by G as its Weyl group; and if G is any finite subgroup of GL(M), then \(C[M]^ G\) is the tensor product of a group algebra and a polynomial algebra (over C) if and only if G is a reflection group and, for some root system, \(M/M^ G\) is isomorphic (as a module over the group induced by G) to a weight lattice (over the Weyl group).