Consider the group algebra $\mathbb{R}[C_3]$，where $\mathbb{R}$ is real field and $C_3$ is $3$-order cyclic group.
It's known $C_3$ can be represented as $\{1,e^{2\pi i/3},e^{4\pi i/3}\}$, I tried to generalize it to the representation of $\mathbb{R}[C_3]$.
If we take linear combinations of $\{1,e^{2\pi i/3},e^{4\pi i/3}\}$ in $\mathbb{R}$, it is not faithful since this is $2$-dimensional but $\mathbb{R}[C_3]$ should be $3D$.
Is it possible to get a faithful representation of $\mathbb{R}[C_3]$ using matrix?
Is there any reference about that?
Let $C_3 = \langle\, g \mid g^3 = 1 \,\rangle$ and consider the representation $\rho \colon C_3 \to GL_3(\mathbb{R})$ defined by $$ \rho \colon g \mapsto \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0 \end{bmatrix}.$$ This is a faithful representation of $C_3$. You might want to read about the regular representation of a group to see why it works in general.