- #1

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[tex]\begin{bmatrix}

\cos \theta & \sin \theta \\

-\sin \theta & \cos \theta

\end{bmatrix}[/tex]

##\theta \in [0,2\pi]##

form a group. This is special orthogonal group ##SO(2)##. However it is possible to diagonalize this matrices and get

[tex]\begin{bmatrix}

e^{i\theta} & 0 \\

0 & e^{-i \theta}

\end{bmatrix}=e^{i \theta}\oplus e^{-i\theta}.[/tex]

It looks like that ##e^{i\theta}## is irreducible representation of ##SO(2)##. However in ##e^{i\theta}## we have complex parameter ##i## and this is unitary group ##U(1)##. Where am I making the mistake?