algebra of invariants

Definition

Let be a finite dimensional vector space (over for simplicity) with a choice of basis and let be a group that acts on via the group representation

acts on by -algebra automorphisms using the formula

This is a group action that respects the algebra structure. The algebra of invariants are the elements of that are invariant under the action, that is,

Motivation

The algebra of invariants is an attempt to understand orbits of a group action. Note that give , seen as a set, we can look at functions out of the set

This gives rise to questions, such as given a group action , is the algebra of invariants Noetherian (which is proved by the Hilbert’s finiteness theorem).