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This question is continuation of an earlier question asked in Matrices which are both unitary and Hermitian

Consider the unitary group $U(n^2)$ and consider the subset $R$ of Hermitian Unitary matrices. We want to find different conjugacy classes of $R$ by the subgroup action of $U(n)\otimes U(n)$, i.e. $R_1$ ~ $R_2$ iff $\exists U\otimes V$ such that $R_1=U\otimes V R_2 U^*\otimes V^*$. Let $\mathcal{R}$ be the set of such conjugacy classes. My question is, how much is known about thee structure of $\mathcal{R}$, i.e. whether this is finite, or finitely generated or isomorphic to some known object. Also, can someone suggest any paper/article discussing the above problem.

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Are you sure $R$ is a subgroup? –  Jason DeVito Apr 13 '12 at 12:33
@Jason DeVito: My mistake. If $R$ is a subgroup, then it has to be commutative, which need not be the case. I have changed the question accordingly considering $R$ as a subset. –  RSG Apr 13 '12 at 15:12
What's the motivation to restrict to $U\otimes U$? –  draks ... Apr 16 '12 at 21:41
@draks Thanks for asking. In quantum mechanics $U\otimes V$ are called local unitary. One use of them is they can not create or destroy quantum entanglement. The $R$ mentioned above is also an observable, and I want to see how many of them are conjugate to each other by the above relation. –  RSG Apr 17 '12 at 6:48
Maybe you are looking for something like this... –  draks ... Apr 17 '12 at 17:45
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