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Let $\phi: \Bbb{Z}/n\Bbb{Z}\to GL_{2}(\Bbb{C})$ be the representation which takes $\bar{m} \to \begin{bmatrix} \cos (\theta) & -\sin (\theta) \\ \sin (\theta) & \cos (\theta) \end{bmatrix}$ where $\theta = \frac{2\pi\ m}{n}$ which is matrix of rotation.

My book says that $W=\Bbb{C}e_1$, where $e_1=(1,0)$, is invariant under $\phi$ but I don't get it. Vectors on X-axis will be rotated. How does rotation does not affect them?

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  • $\begingroup$ It seems that your book is wrong $\endgroup$ – Omnomnomnom Sep 25 '15 at 19:48
  • $\begingroup$ it is on page 15 of steinbergs' Rep theory of Finite groups $\endgroup$ – Departed Sep 26 '15 at 5:16

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