I've recently been studying polynomial functions which remain invariant under linear transformations.
I have considered the ring $\mathbb{C}[x_1,...,x_n]$, and let a group of matrices $G$ act on $\mathbb{C}[x_1,...,x_n]$ like so: $$A \cdot f(\textbf{x}) = f(A \textbf{x})$$ where we are to think of $\textbf{x}$ as a column vector in the $x_i$.
Now, all this does is apply linear trasformations to the variables of $f$.
Then, the invariant subring is $\mathbb{C}[x_1,...,x_n]^G$, which is the set of all $f \in \mathbb{C}[x_1,...,x_n]$ such that $f(x_1,...,x_n) = f(A\textbf{x})$ for all $A \in G$.
Now, how does representation theory come in to this? There's groups of matrics, group actions and so on, surely there must be a link somewhere?