1
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

Since $\mathbb{Z}/3\mathbb{Z}$ is cyclic it is sufficient to diagonalize the matrix $\rho (1)$.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.