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Let $V=\mathbb{C^2}$ be the standard representation of $SO_2$

Decompose $V$ into irreducible representations

The standard unit vectors of $\mathbb{C^2}$ are $e_1$ and $e_2$

I am not sure how to use these to decompose $V$, does it have something to do with finding eigenvectors?

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  • $\begingroup$ You mean real irreducible representation right? And $\mathrm{SO}(2)$, with real matrices? $\endgroup$ – arctic tern May 2 '16 at 2:25
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A way to do this is to use the Lie algebra: if $$ J = \pmatrix{ 0& -1\\1 & 0 } = {d\over d \theta }{\Large|}_{\theta = 0 }\pmatrix{ \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta},$$ then $$J^2 + 1 = 0.$$ Diagonlize $J$: then the rotation $\exp J\theta $ will act on (stabilize) the eigen-spaces of $J$.

  • btw: $ \exp J\theta$ acts by multiplication by $e^{i\theta}= \cos \theta + i \sin \theta$ on one eigen-space, and by multiplication by $e^{-i\theta}= \cos \theta - i \sin \theta$ on the other.
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I assume you mean $\mathrm{SO}(2)$ and not $\mathrm{SO}(2,\mathbb{C})$. Notice real matrices commute with multiplication by the imaginary unit $i$, and the most obvious real irreducible representation of $\mathrm{SO}(2)$ is $\mathbb{R}^2$. It should be clear there is a copy of $\mathbb{R}^2$ within $\mathbb{C}^2$. Can you find another copy? (Use multiplication by $i$.)

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