Weingarten formula

(1)

where for $\sigma \in S_n$

$\begin{equation} \operatorname{Wg}_{N}(\sigma) =\sum_{\lambda\vdash n\atop \ell(\lambda)\le N} \frac{s^2_{\lambda}(\delta_{\boldsymbol{\cdot},1})}{s_{\lambda}(N)} \chi^{\lambda}(\sigma) . \end{equation}$ (2)

Schur polynomials $s_\lambda$ are implemeted e.g. in SageMath, and characters $\chi^{\lambda}(\sigma)$ can be obtained either using charater_table() or via Frobenius characteristic map from Schurs.

$\begin{equation} \int_{U(N)}dU \left( \prod_{k=1}^n U_{i_k a_k} \right)\left( \prod_{m=1}^p U_{j_m b_m}^{*} \right) =0 \end{equation}$ (3)

for $p \ne n$ because the Haar measure is invariant under the central $U(1) \subset U(N)$ ,

$\begin{equation} U \to e^{i \theta} U, \qquad dU \to dU, \end{equation}$ (4)

while under this phase shift

$\begin{equation} U_{ia} \to e^{i \theta} U_{ia}, \qquad {U_{j b}^{*}} \to e^{-i\theta}{U_{j b}^{ *}}, \end{equation}$ (5)

the integrand picks up a factor $e^{i\theta(n - p)}$ . Hence for $n \ne p$

$\begin{equation} \int_{U(N)}dU \left(\prod_{k=1}^n U_{i_k a_k}\right) \left(\prod_{m=1}^p {U_{j_m b_m}^{*}}\right) = e^{i\theta(n-p)} I \quad\forall \theta \Longrightarrow I=0. \end{equation}$ (6)

References

[CŚ06]

B. Collins and P. Śniady, “Integration with respect to the haar measure on unitary, orthogonal and symplectic group,” Comm. Math. Phys., vol. 264, no. 3, pp. 773–795, 2006, doi: 10.1007/s00220-006-1554-3. arXiv: math-ph/0402073.

[Col03]

B. Collins, “[],” Int. Math. Res. Notices, vol. 2003, no. 17, p. 953, 2003, doi: 10.1155/s107379280320917x.

[Wei78]

D. Weingarten, “Asymptotic behavior of group integrals in the limit of infinite rank,” J. Math. Phys., vol. 19, no. 5, pp. 999–1001, 1978, doi: 10.1063/1.523807.