Invariant subring for a linear representation

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

 * Invariant subring for an algebraic group action
 * Invariant subring for an action on an algebra