I am trying to solve the problem:
Decompose the standard representation of the cyclic group $C_{n}$ in $\mathbb{R}^{2}$ by rotations into a direct sum of irreducible representations.
What I have tried:
Taking $x$ to be a generator, I define $\rho:C_{n}\to\mathbb{R}^{2}$, $x^{k}\mapsto\left[\begin{array}{cc}cos(2k\pi/n) & -sin(2k\pi/n)\\ sin(2k\pi/n) & cos(2k\pi/n)\\\end{array}\right]$
But I can't think of any $C_{n}$-invariant subspace of $\mathbb{R}^{2}$. The $x$-axis, $\mathbb{R}\times \{0\}$ might be $H$-invariant for a subgroup $H\leq C_{n}$ which might give me an induction/restriction relationship. Anyhow that doesn't seem to go anywhere.
How can I find a $C_{n}$-invariant subspace $W\subset V$.
Or does this mean that $\rho$ is an irreducible representation?