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Let C be the complex field, and $G$ be a cyclic group generated by $a$. The group ring $C[G]$ is a $C[G]$-module (over itself) with the module action $C[G]\times C[G] \rightarrow C[G]$ the same as the group ring action.

The problem states to give a decomposition of C[G] as a direct sum of simple modules of the form $Ce_{\chi_m}$ where $$e_{\chi_m}=\frac{1}{n}\sum_{a^k \in\langle a\rangle} \exp\left(2\pi i \frac{m}{n} k\right) a^{-k}$$

I am guessing that $$C[G] \cong \bigoplus ^{n-1}_{j=0} Ce_{\chi_j}$$

I've showed that $Ce_{\chi_m}$ is a submodule of rank 1 and that $e_{\chi_m}$ is idempotent but I'm having some difficulty showing how the $e_{\chi_m}$'s are linearly independent.

I am trying to show if they are linearly independent, since if so then it is clear we have the direct sum. Perhaps I should try another method?

edit: Originally the question was phrased with $e_{\chi_m} = \frac{1}{n}\sum_{a^k \in\langle a\rangle} \chi_m(a^k) a^{-k}$, but I found $$\chi(a^k) = \exp\left(2\pi i \frac{m}{n} k\right)$$ for integers $m$ to be the only possible morphisms of the form $\chi : G \rightarrow C^\times$

edit2: title

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    $\begingroup$ Hint: the Vandermonde determinant. $\endgroup$ – Orat Apr 11 at 11:14
  • $\begingroup$ Thanks @Orat. That is basically what I need $\endgroup$ – CloudIcarus Apr 12 at 2:06

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