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I'm reading a bit about the Wigner D matrix, defined by

[itex]\mathscr{D}\left(\hat{n},\phi \right) = \exp[-\frac{i \phi}{\hbar}\vec{J}\cdot \hat{n}][/itex].

Now I'm wondering : is the map [itex] \pi : \text{SO(3)} \to \text{GL}\left( \mathscr{H} \right) [/itex] given by [itex] R\left(\hat{n},\phi \right) \mapsto \mathscr{D}\left(\hat{n},\phi \right)[/itex] a representation of SO(3) on some Hilbert space [itex]\mathscr{H}[/itex]?

I would say yes, but I have no clue on how to prove that this map is a homomorphism so that it is indeed a representation. Further, if it is a representation, is it reducible or irreducible?

Thanks for any help

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# Is the wigner D function a representation of SO(3)?

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