Weingarten function for unitary group

Question

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?

Answer 1

I would recommend Elementary derivation of Weingarten functions of classical Lie groups 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.