Is there any more or less efficient way to integrate a function (not necessarily a polynomial) over $U(d)$?
Performing integration over $U(d)$
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integration
lie-groups
rational-functions
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0$U(d)$ is an element of $\mathcal{U}(d)$, the $d$-dimensional unitary matrix group? Or is $d$ a variable, like $U(d)=e^{dH}$, with skew-hermitian $H$? – 2012-04-03
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0By $U(d)$ I mean the group of unitary $d\times d$ matrices. – 2012-04-04
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0Could you please provide an expression of how your integrals would look like? – 2012-04-04
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2I can try. I need to perform an integral of the type $\int g_W(r,U) f(r)drdU$, where $dr$ is a Lebesgue measure on a cube $[0,1]^{d-1}$, $dU$ is a Haar measure on $U(d)$, $f(r)$ is known and $g_W(r,U)=\frac{1}{2}(1-\frac{Tr(W U diag\{r1,...,r_d\} U^{\dagger})}{|Tr(W U diag\{r1,...,r_d\} U^{\dagger})|})$, with some constant fixed matrix $W$. – 2012-04-05
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0Wow! That's far more info than in your original post... Maybe you should update your post! I don't think I can help you here, but I'm curious about the answer. Good luck! – 2012-04-05