Chose e.g. $e_1 = (1,0,0,...,0)$. What does the set $\{D^m(g)e_1, g \in SO(3) \}$ look like? ($D^m$ is an m-dimensional irrep of SO(3), $m$ > 3)
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There is no way to actually give the answer explicitly for your $e_1$ as that is not a well-defined vector given just some $n$-dimensional representation. Note that the set will "almost" be everything, except that you will be missing some scalars. – Tobias Kildetoft Aug 03 '16 at 14:42
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Hi Tobias, thanks for the comment, sorry for being unclear. I try to be more specific: $D^m$ is the Wigner matrix acting on spherical harmonics. (So $R^m$ is the space of coefficients vectors where the basis vectors are spherical harmonics.) – alain Aug 03 '16 at 14:53
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..and trying to be more precise: 'look like' in the sense of: for the 'direct' representation of SO(3) (m=3) the orbits are 2D spheres. Is there such a 'intuitive' manifold for the general case? – alain Aug 03 '16 at 15:00
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@TobiasKildetoft Why do you think that the orbit will "almost" be everything? – Peter Franek Aug 15 '16 at 19:32