Haar integral of rational function of unitaries

Question

I'm trying to compute the following Haar integral over the unitary group: $$ \int\limits_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l=1}^d u_{ik}\overline{u_{il}}c_{kl}}dU. $$ Is there anything known about the value of such integrals? I haven't been able to find anything that studies the integrals of either non polynomial functions or non class functions of unitaries.

Answer 1

There exist no closed-form expressions for arbitrary $d$ for the integral over the unitary group $\mathbb{U}(d)$ of a rational function of the matrix elements.
There are asymptotic results for large $d$, see for example J. Math. Phys. 37, 4904 (1996). The leading order term for $\text{tr}\,C$ of order $d$ is $$\int\limits_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l}u_{ik}\overline{u_{il}}c_{kl}}dU=\frac{d}{\text{tr}\,C}+{\cal O}(1/d),$$ independent of the index $i$.

Alternatively, if $d$ is not large but the matric $C$ is close to the unit matrix, $c_{kl}=\delta_{kl}+\epsilon_{kl}$, one can expand $$ \begin{split} \int\limits_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l}u_{ik}\overline{u_{il}}c_{kl}}dU & =1-\int\limits_{\mathbb{U}(d)}\sum_{k,l}u_{ik}\overline{u_{il}}\epsilon_{kl}\,dU+{\cal O}(\epsilon^2)\\ & =1-\frac{1}{d}\sum_{k}\epsilon_{kk}+{\cal O}(\epsilon^2). \end{split}$$