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Consider the component-wise action of the group $SO(p)\times SO(q)$ in the tensor product of two real vector spaces $S^2(R^p)\otimes R^q$. How to parametrize orbits of this action ? For $q=1$ we obtain a nice answer: the parameters are eigenvalues of the bilinear form.

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what's $S^2(R^p)$ ? – mercio Nov 7 '12 at 15:32
$R^p$ - the standard p-dimensional inner product space, $S^2 (R^P)$ - the second symmetric power of $R^p$, or the space of symmetric bilinear forms of $R^p$, here the duality doesn't matter. – Vlambda Nov 7 '12 at 16:35

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