My professor wrote in his notes that a one-dimensional representation of the cyclic group $C_n$ with $a$ as a generator could be written as follows.
$\phi_m(a^k)=e^{2\pi imk/n}$ for some $k \in \{0,1,2,…,n-1\}$ and $m$ is also some integer (not further specified in notes). This is quite clear for me, we can easily check that this is a representation since it is a homomorphism.
But now he defines a two-dimensional representation as the direct sum $\phi_1 \oplus \phi_{-1}$ given by the following matrix:
$(\phi_1 \oplus \phi_{-1})(a^k) = diag(e^{2\pi ik/n},e^{-2\pi ik/n})$ so a diagonal matrix with the one-dimensional representations on the diagonal.
In representation theory we always want a homomorphism given by:
$\phi : G \rightarrow GL(V) \cong Mat(n,K)$ where $V$ is the representation space over a field $K$ and because we are dealing with finite vector spaces we can always identify a linear map $\phi \in GL(V)$ with a matrix $\textbf{given a basis}$ to get $\phi \in Mat(n,K)$.
My question is: with respect to what basis is this matrix?
Best Answer
It doesn't matter with respect to what basis it is. It doesn't even matter with respect to what vector space it is. For instance, you could have a representation of $C_n$ on the set of all complex-valued functions on $S_2$, and the basis could consist of the functions given by the symmetric and the antisymmetric character on $S_2$. What matters is that the matrices have the same multiplication law as the group elements; the underlying vector space is arbitrary and ultimately irrelevant.
Having said that, I suspect that your professor perhaps had the canonical basis for $\mathbb C^2$ in mind.