TITLE: Representation theory notation doubt
QUESTION [1 upvotes]: So I've been reading a little bit about representation theory from the book "Linear Representations of finite groups" and i have a question.
When the author writes , 'Let n be the order of G, and let V be a vector space of dimension n,
with a basis $(e_t)_ {t \in G}$ indexed by the elements t of G." Im not quite sure how to look at this , i dont think this is supposed to be $\mathbb{R} ^n$ , i guess my doubt is that i dont understand what he means with that basis notation. So any help is appreciated thanks.

REPLY [2 votes]: Just to add to G. Chiusole’s excellent answer, I wanted to give a (tiny) example.
Let $G = \mathbb{Z}/3\mathbb{Z} = \{0,1,2\}$. Note that $G$ has three elements, so we have three vectors in $\mathbb{C}^3$:
$$e_0 = \begin{bmatrix} 1 \\ 0 \\ 0\end{bmatrix}, \qquad e_1 = \begin{bmatrix} 0 \\ 1 \\ 0\end{bmatrix}, \qquad e_2 = \begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix}.$$
I know that $0$ is the identity element of $G$, so my representation of $G$ should send $0$ to
$$\begin{bmatrix} 1&0&0\\0&1&0\\0&0&1 \end{bmatrix}.$$
Now, since 
$$\begin{cases} 1+0=1 \\ 1+1=2 \\ 1+2=0\end{cases}$$
my representation should send $1$ to a matrix $A$ such that
$$\begin{cases} Ae_0 = e_1 \\ Ae_1 = e_2 \\ Ae_2 = e_0\end{cases}$$
Thus, we need that
$$A = \begin{bmatrix} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{bmatrix}.$$
Lastly, since 
$$\begin{cases} 2+0=2 \\ 2+1=0 \\ 2+2=1\end{cases}$$
my representation should send $2$ to a matrix $B$ such that
$$\begin{cases} Be_0 = e_2 \\ Be_1 = e_0 \\ Be_2 = e_1\end{cases}$$
Thus, we need that
$$B = \begin{bmatrix} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{bmatrix}.$$
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