What are the irreducible representations of the cyclic group $C_n$ over a real vector space $V$?

It suffices just to consider a linear transformation $f$ such that $f^n=id$ and require $V$ to have no proper subspace invariant under $f$. But I still don't have a picture of what's going on.

Let $\rho : C_n \to \mathrm{GL}(V)$ be a representation. Equivalently, let $f : V \to V$ be an automorphism such that $f^n = \operatorname{id}_V$. Then $f$ is a root of the polynomial $X^n - 1$. This polynomial splits as a product of linear terms over $\mathbb{C}$. Therefore $f$ is diagonalizable over $\mathbb{C}$. It follows that $V$ splits as a direct sum of the sub-representations (over $\mathbb{C}$):

$$V \otimes_{\mathbb{R}} \mathbb{C} = \bigoplus_{i=1}^k \ker(f - \lambda_i \operatorname{id}_V),$$

where the $\lambda_i$ are the complex eigenvalues of $f$. These are $n$th roots of unity since $f^n = \operatorname{id}$.

The root $\lambda_i$ is either:

• Real, in which case $\ker(f - \lambda_i \operatorname{id})$ is also a sub-representation. It splits as a direct sum of one-dimensional representations (simply choose a basis).
• Or it comes in a pair of complex conjugate numbers. Indeed, let $A \in M_d(\mathbb{R})$ be the matrix associated to $f$. Then $Av = \lambda_i v$ for a nonzero $v$ in the eigenspace, and therefore: $$\overline{Av} = \overline{\lambda_i v} \implies A \bar{v} = \bar{\lambda}_i \bar{v},$$ and therefore $\bar{\lambda}_i$ is also an eigenvalue of $A$.

Now you can pair up $\lambda = e^{2ik\pi/n}$ and $\bar{\lambda} = e^{-2ik\pi/n}$. Let $\theta = 2k\pi/n$, such that $n \theta \equiv 0 \pmod{2\pi}$. The two following matrices are similar: $$\begin{pmatrix} e^{i \theta} & 0 \\ 0 & e^{-i\theta} \end{pmatrix} \sim \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$

Conclusion: The irreducible real representations of $C_n$ are either

• 1-dimensional, with matrix a real $n$th root of unity;
• 2-dimensional, with matrix $\left(\begin{smallmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{smallmatrix}\right)$ where $n\theta \equiv 0 \pmod{2\pi}$ but $\sin\theta \neq 0$.