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A finite-dimensional unitary representation $U$ of a group $G$ can be decomposed into a direct sum of irreducible representations $(U_i)_{1\leq i\leq n}$: $$U=\oplus_{i=1}^{n} \, U_i .$$ According to this result, as $\chi_m(t)=e^{imt}$ is a finite-dimensional unitary representation of $\mathbb R$. How can decompose this representation ?

Thank you in advance

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    $\begingroup$ It is already irreducible, as it is one-dimensional. You can't decompose it any more! $\endgroup$ – Giuseppe Negro Jan 25 '18 at 11:27
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If you are talking about the group $\left ( \mathbb{R},+ \right )$, then your representation is automatically one (complex) dimensional, which implies it is not decomposable (since it does not admit any direct sum decomposition).

Also, it is trivially irreducible since $\mathbb{C}$ only has two subspaces and they are $\varnothing$ and $\mathbb{C}$ itself.

In general, irreducibility and indecomposability are different, though in our case they are the same.

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