How to get group representation of the cyclic group $C_3 = \{1,u,u^2\}$ where $u = e^{2\pi i / 3}$ I grasped the concept of group representation, but I am not sure how to get group representation of the cyclic group $C_3 = \{1,u,u^2\}$ where $u = e^{2\pi i / 3}$. Let's say we want to get a representation $\rho$ on $\textbf{C}^2$. How does one do this?
 A: To rephrase your question, you are looking for a homomorphism $\rho:C_3\to\mathrm{GL}_2(\mathbb{C})$.
There is always the trivial representation, which sends everything to the identity matrix. 
Another way, which is less trivial, is to find a representation $\sigma$ of $C_3$ on $\mathbb{C}$ (hint: $\mathrm{GL}_1(\mathbb{C})=\mathbb{C}^\times$, the multiplicative group of non-zero complex numbers), then construct a representation $\rho$ on $\mathbb{C}^2$ by making elements of $C_3$ act "coordinate-wise" on elements of $\mathbb{C}^2$.
A: Your should know that all irreducible representations of  the abelian group $C_{3}$ are one-dimensional.
There are three of them, given by
$$
C_{3} \to \operatorname{GL}_{1}(\Bbb{C}) \cong \Bbb{C}^{\times}, \qquad u \mapsto u^{k}, 
$$
for $k = 0, 1, 2$.
So to get a two-dimensional representation, just combine any two of these. Among the various choices, the one
$$
C_{3} \to \operatorname{GL}_{2}(\Bbb{C}), \qquad u \mapsto 
\begin{bmatrix}u&0\\0&u^{2}\end{bmatrix}
$$
is probably particularly interesting, as it can be realized over the reals:
$$
C_{3} \to \operatorname{GL}_{2}(\Bbb{R}), \qquad u \mapsto 
\begin{bmatrix}0&-1\\1&-1\end{bmatrix}.
$$
(But see Jyrki Lahtonen's comment below for the much more natural representation:-) in terms of rotations.)
