I am asked to determine all irreducible $FG$-modules of dimension $n$ for $G=C_k$, the cyclic group of order $k\in \mathbb N$ in the following cases:
- $n=1, F= \mathbb R$
- $n=2, F= \mathbb R$
- $n=1, F= \mathbb C$
- $n=2, F= \mathbb C$
I honestly do not know where to start. This seems like a classification question, can someone help me to get started?
This follows on from the discussion in the comments.
Let $V_k$ be the $2$-dimensional $\mathbb{R}C_k$-representation that you described. We shall show that $V_k$ is irreducible when $k > 2$, and reducible when $k = 1,2$
Suppose that $k>2$ and suppose, for the sake of contradiction, that $V$ is reducible, that is it contains some non-zero proper subrepresentation $U \subseteq V$. Since $V$ is $2$-dimensional, that forces $U$ to be one-dimensional. In other words any non-zero $u \in U$ is simultaneously an eigenvector for the action of $C_k$ (or alternatively an eigenvector for its generator since it is cyclic). But this generator has no eigenvectors when $k > 2$, and so we have reached our contradiction.
When $k = 1,2$ the vector $\begin{pmatrix} 1 \\ 0\end{pmatrix}$ is an eigenvector of the generator of $C_k$, and so generates a non-trivial one-dimensional submodule of $V_k$.
For an extension, why does this argument fail if we replace $\mathbb{R}$ with $\mathbb{C}$?