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 $\rho (1)$.

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

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