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It is mentioned in Wiki that the spaces $\mathcal{H}_{k}$ of spherical harmonics of degree $k$ give ALL the irreducible representations of $SO(3)$. Could anyone tell me where can I find the proof? Thanks!

EDIT: I am seeking for an elementary proof that dose not require too many big machineries in representation theory.

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    @ramanujan_dirac: Dear ramanujan_dirac, I don't know off the top of my head, and I'm not sure I'm the right person to really answer this kind of question. Regards,2014-06-06

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The proof is rather simple, just calculate characters (for rotations around OZ, since the axis does not affect the character) , and, using orthogonality theorem, note that all Fourier series coefficient for any other character are zero. But functions $\cos (l*\phi) (l - n)$, (where $n$ is an integer) form a complete set on $<0, \pi>$, so there are no more irreducible, unequivocal representations of $SO(3)$.

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    Characters of these representations for SO(3) are dependent only on one rotation parameter. This is not true for SO(4). You can find a basis of Lie algebra A1..A3, B1..B3 such that: [Ai, Aj] = const*Ak, [Bi, Bj] = const*Bk, [Ai,Bj] = 0 for any i, j. So$SO(4)$is isomorphic to SO(3) x SO(3). That means each conjugancy class is defined by two parameters, which makes further calculations much harder. Moreover, this method with Fourier series works for SO(3), only because the characters are some simple trigonometric functions, dependent on dimmension. In the general case it may not work.2013-08-31