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Let $SO(3)$ act on the space of $3 \times 3$ real matrices by conjugation. How can I decompose the space of matrices into the sum on minimal invariant subspaces and figure out what they are isomorphic to?

I am familiar with the irreducible representations of $SU(2)$ and how they give the irreducible representations of $SO(3)$. I don't see how to relate this notion to the minimal invariant subspaces.

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    Can you compute the character of this representation?2012-01-09
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    Conjugation maps symmetric matrices into symmetric, anti-symmetric into anti-symmetric, preserves trace, determinant. This should be enough to get you going...2012-01-09
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    Add to Sasaha's hint: check that conjugation (or transpose) commutes with the action of SO(3).2012-01-09

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