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Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...

Given two lists $i$ and $j$ of $2n$ positive integers less than $N$, Collins and Sniady have computed, in Integration with respect to the Haar measure on unitary, orthogonal and symplectic group (see ...

In a physics paper I found a very complicated Gaussian matrix model: $$ Z = \int \frac{d\mu}{(2\pi)^n} \frac{d\nu}{(2\pi)^n} \frac{ \prod_{i < j}\left[2 \sinh \frac{\mu_i - \mu_j}{2} \right]^2 \...

It is known how to calculate the integral of elements of $N\times N$ Haar random unitaries using the Weingarten function: $$\int \prod_{k=1}^n U_{i_kj_k} U_{m_kr_k}^* \mathrm d U = \sum_{\sigma,\tau} \...