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Coming from physics I have come across the following integral over a haar measure (for $U$ unitary as an example) for something I am trying to determine for my work

$\int_{\mathcal{U}(d)} \frac{\mathrm{Tr}(XUAU^{\dagger} \Pi)^{2}}{\mathrm{Tr}(UAU^{\dagger} \Pi)}dU$

where $\Pi$ is idempotent (and if relevant commutes with $X$) and $A$ can be assumed to be a rank 1, trace 1 matrix.

I understand that we can use Weingarten calculus to compute these kinds of integrals when they are a polynomial of $U$ and $U^{\dagger}$, but I haven't seen anything for when there are quotients involved.

Is there some known way to compute these kinds of integrals?

I have come across Haar integral of rational function of unitaries which is similar but only has the denominator, but I am wondering if similar techniques are known when there are additional terms of $U$ in the integral

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  • $\begingroup$ if I may assume that $A$ is Hermitian, the integral can be rewritten as $$\int \left(\sum_n y_n |U_{n1}|^2\right)\left(\sum_n z_n|U_{n1}|^2\right)^{-1}\,dU$$ for some set of coefficients $y_n,z_n$; there is no closed form answer; for large $d$ you could take independent normal distributions for the $U_{n1}$ elements, but that's about it. $\endgroup$
    – Carlo Beenakker
    Commented Oct 6, 2022 at 13:12

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