Invariant subring for a linear representation

This term makes sense in the context of a linear representation of a group, viz an action of the group as linear automorphisms of a vector space

Definition

Let $G$ be a group and $\rho$ a linear representation of $G$ over a field $k$ , viz $\rho :G\to GL(V)$ is a homomorphism. Then, the invariant subring for the linear representation $\rho$ is defined as the subring of $k[V]$ comprising those algebraic functions that are invariant under the action of $G$ .

Here, when $V=k^{n}$ , $k[V]$ is the polynomial ring $k[x_{1},x_{2},\ldots ,x_{n}]$ . Thus the invariant subring is simply a subring of the polynomial ring comprising those polynomials that are invariant under the group action.

Generalizations