Given two lists $i$ and $j$ of $2n$ positive integers less than $N$, Collins and Sniady have computed (see also this paper 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 $SO(N)$?

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 $SO(N)$?

Given two lists $i$ and $j$ of $2n$ positive integers less than $N$, Collins and Sniady have computed 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 $SO(N)$?

Given two lists $i$ and $j$ of $2n$ positive integers less than $N$, Collins and Sniady have computed (see also this paper 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 $SO(N)$?