# write representation as sum of irreducible representations

Given the representation $\rho: \mathbb{Z}/3\mathbb{Z} \rightarrow GL_2(\mathbb{C})$ by $1\rightarrow \left( \begin{array}{ccc} -1 & -1 \\ 1 & 0\\ \end{array} \right)$. I have to write this representation as a sum irreducible representation. I tried a lot to figure this out, but I just dont ´see´ it. I need help. Thanks.

Since $\mathbb{Z}/3\mathbb{Z}$ is cyclic it is sufficient to diagonalize the matrix $\rho (1)$.
• could you explain why it is sufficient to diagonalize $\rho(1)$? – Badshah May 10 '13 at 15:41
• Then the matrix $\rho(2)=\rho(1)^2$ is also diagonal and you obtain a decomposition into two 1-dimensional representations. – Boris Novikov May 10 '13 at 15:45