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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 also On some properties of orthogonal Weingarten functions by Collins and Matsumoto), the integral over the orthogonal group, $$ \int_{O(N)} \prod_{k=1}^{2n}u_{i_kj_k}du=\sum_{\sigma,\tau}\Delta_\sigma(i)\Delta_\tau(j) {\rm Wg}_N(\sigma^{-1}\tau),\qquad (1)$$ where the sum is over matchings, $\Delta_\sigma(i)=1$ if and only if the sequence $i$ satisfies the matching $\sigma$ and ${\rm Wg}_N$ is called the Weingarten function.

This implies for instance that $\int_{O(N)} u_{11}u_{22}du=0$ because the list $(1,2)$ does not match.

On the other hand, we know that a matrix from $SO(2)$ is of the form $u=\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{pmatrix}$ so clearly we have $\int_{SO(2)} u_{11}u_{22}du=1/2$. This shows that the $SO(N)$ result can be quite different from the $O(N)$ one.

Is there a general theory of integrals like (1) over $SO(N)$?

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    $\begingroup$ you can of course multiply the integrand by the factor $1+{\rm det}\,u$ to constrain the average over $O(N)$ to $SO(N)$; I have used this approach in App.B of arXiv:1012.0932 for a particular problem in the context of topological superconductivity. $\endgroup$ Jun 26 '20 at 20:36
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That's a good question. This is not quite an answer but a bit long for a comment.

A quick remark is that for $N$ odd a Haar-random element of $O(N)$ can be obtained as $\epsilon U$ where $\epsilon=\pm1$ with equal probability, and $U$ is a Haar-random element of $SO(N)$. So if your monomial has an even number of factors the integrals over $O(N)$ and $SO(N)$ coincide and thus Weingarten calculus is applicable. This is of course because for $N$ odd $-I$ has determinant $-1$ and is in the center of $O(N)$. I don't know if there is a similar trick for $N$ even.

I said it is a good question because, when looking at the vast probability/representation theory literature, I didn't see much as far an analogue of Weingarten calculus for special groups. Even the work of Chatterjee (and Basu and Ganguly,...) on $SO(N)$ lattice gauge theories does not seem to use Weingarten calculus. So for $SO(N)$, my answer to the question is: I don't know. However, for $SU(N)$ there is a combinatorial calculus. It is explained in my two answers to

How to constructively/combinatorially prove Schur-Weyl duality?

This technique was worked out explicitly by Creutz but it has its roots in the work of Clebsch and Hilbert in invariant theory. See for example, the averaging operator $[\cdot]$ used by Hilbert on p. 523 of "Ueber die Theorie der algebraischen Formen" is basically the same as Creutz's formula for $SU(2)$. Also note that if a combinatorial Weingarten-like calculus for $SO(N)$ is perhaps missing, there is at least an Euler angle parametrization due to Hurwitz (see this review by Diaconis and Forrester).

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  • $\begingroup$ Thank you for your input. The idea of working with explicit Euler angles seems scary. Let me point out that it is easy to derive the Weingarten function for $O(N)$ using the theory of zonal polynomials, as I show here: arxiv.org/abs/1406.2182. Zonal polynomials appear because of the relation $\int_{O(N)}s_{2\lambda}(Au)du=Z_\lambda(A^TA)/Z_\lambda(1)$, where $s$ is a Schur function. So a Weingarten calculus for $SO(N)$ might follow from the integral $\int_{SO(N)}s_{\lambda}(Au)du$. Is this integral known? $\endgroup$
    – Marcel
    Jun 26 '20 at 20:04

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