I can see how to go about proving reducible representations by finding projection matrices, but I don't know how to prove the inexistence of these projectors to prove a representation is irreducible. Is there a way to do this? I'm working on simply proving the $D(e)=1, D(a)=e^{(2i\pi/3)},D(b)=e^{(4i\pi/3)}$ an irreducible representation of $Z_3$, given by the multiplication table:
$$ \begin{array}{c|c|c|c|} \backslash & e & a & b \\ \hline e & e & a & b \\ \hline a & a & b & e \\ \hline b & b & e & a \\ \hline \end{array} $$
Thank you!
A commutative group will always contain $\vert G \vert$ 1-dimensional irreducible representations since $\sum_{i=1}^{\vert G\vert}1^2 = \vert G\vert$ this is due to every conjugacy class being unique as consequence of commutativity $a^{-1}b{a} = b$ thus the action by conjugation only yields $b$ as the unique member of the conjugacy class $b$.
This is strongly connected to character theory and a good article to start with is the associated to Dirichlet characters of multiplicative groups of integers mod $n$. https://en.wikipedia.org/wiki/Dirichlet_character#A_few_character_tables