Studying integration over unitary group I came across this function, the Weingarten function Wg, such that
$$ \int\_{\mathcal{U}(N)} \prod\_{k=1}^{n} U\_{i\_kj\_k}
U^\*\_{m\_k r\_k} dU=\sum\_{\tau,\sigma\in S\_n} {\rm
Wg}^U(\tau^{-1}\sigma)\prod\_{k=1}^n \delta\_{i\_k,\tau(m\_k)}\delta\_{j\_k,\sigma(r\_k)}.$$ I understand the structure of this equation, the $m$'s must be a permutation of the $i$'s, and likewise the $r$'s with the $j$'s. However, the function itself is complicated, 
$$ {\rm
Wg}(\tau^{-1}\sigma)=\frac{1}{n!^2}\sum\_{\substack{\lambda\vdash n\\\ell(\lambda)\leq N}}
\frac{d\_\lambda^2}{s\_\lambda(1^N)}\chi\_\lambda(\tau^{-1}\sigma),$$ where $s\_\lambda$ are Schur functions, $d\_\lambda$ is the dimension of a irreducible representation of the permutation group labelled by $\lambda$ and $\chi$ are the characters of the irreps of this group. The derivations I have seen of this result are also complicated.


Can anyone offer some insight into this function and how it is derived?



Weingarten function for unitary group
I would recommend [Elementary derivation of Weingarten functions of classical Lie groups](http://arxiv.org/abs/1406.2182) by Marcel Novaes (2015).



> 
> Previous works where Weingarten functions were obtained were based
>  either on representation theory and Schur-Weyl duality, the theory of
>  Gelfand pairs, or Jucys-Murphy elements. In contrast, we here derive
>  Weingarten functions for the classical compact groups by means of some
>  elementary direct calculations (although we rely on some classical
>  results that can, of course, be interpreted very naturally in the
>  light of those theories).
> 
> 
> The idea consists of five steps: 1) write the integrand as the
>  derivative of a power sum function; 2) change basis from power sums to
>  Schur functions; 3) perform the group integral; 4) revert back to
>  power sums; 5) take the derivative to arrive at the result.
> 
> 
> 


