# Decomposition of $C[G]$ as $C[G]$-module into direct sum of submodules of the form $Ce_{\chi}$

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

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