Rings of Invariants of Finite Groups

We shall prove the fundamental results of Hilbert and Noether for invariant subrings of finite subgroups of the general linear groups in the non-modular case, i.e. when the field has characteristic zero or coprime with the order of the group. We will also derive the Molien’s formula for the Hilbert series of the ring of invariants. We will show, through examples, that the Molien’s formula helps us to see when to stop computing the invariants. We will calculate, quite explicitly, the ring of invariants of the dihedral and icosahedron groups. Finally we will present a rapid treatment of Cohen-Macaulay graded rings. The Cohen-Macaulay property is one of the crucial properties of the rings of invariants. We will show how this property helps us in presenting the ring of invariants in an economical manner via the so called Hironaka decomposition.