When trying to build a dual formulation for lattice gauge theories using Weingarten integration I am getting sums of the kind

$$I^{m, n}_{\mu, \nu} (\sigma, \tau) = \sum_{\pi \in S_n} \chi_\mu (\pi \sigma) \chi_\nu(\pi \tau), $$

where $\sigma$ and $\tau$ are some fixed permutations in $S_m$ ($m \geq n$), $\mu$ and $\nu$ are some irreducible representations of $S_m$ defined by the partitions of $m$, and $S_n$ is a subgroup of $S_m$ leaving the last $m - n$ elements in place.

Now when $m = n$ the sum is trivial using the orthogonality rule for irreducible representations. Also when $\sigma$ and $\tau$ are in $S_n$, I can just use the Littlewood-Richardson rule to expand restrictions of $\mu$ and $\nu$ to $S_n$ into irreducible representations of $S_n$ and get the result. But can something be done in the general case?

I guess the answer cannot be written in terms of $S_m$ characters of $\sigma$ and $\tau$ since fixing the $S_n$ subgroup breaks the permutation symmetry -- in particular $$ I^{m, n}_{\mu, \nu} (\rho^{-1} \sigma \rho, \rho^{-1} \tau \rho) = \Sigma^{m, n}_{\mu, \nu} (\sigma, \tau) $$

does not necessary hold for every $\rho \in S_m$, (just for $\rho$ in $S_n$). Still maybe there is some way to obtain the sum?

One possibility I am thinking of is taking the representation matrices for $\pi$, $\sigma$ and $\tau$ as $\Pi_\mu$, $\Sigma_\mu$, $T_\mu$ and rewrite

$$I^{m, n}_{\mu, \nu} (\sigma, \tau) = \sum_{\pi \in S_n} {\Pi_\mu}_{i j} {\Sigma_\mu}_{j i} {\Pi_\nu}_{k l} {T_\nu}_{l k}, $$ where $i$, $j$ enumerate Young tables of shape $\mu$ and $k$, $l$ enumerate Young tables of shape $\nu$.

After that if I could somehow express ${\Pi_\mu}_{i j}$ in terms of the matrix elements of irreducible representations of $S_n$, I would be able to use the orthogonality property for the matrix elements and get an answer in terms of matrix elements of $\sigma$ and $\tau$. But this would require some analogue of Littlewood-Richardson rule for matrix elements. (There should be a basis in which $\Pi_\mu$ is a block diagonal matrix, built from irreducible representation matrices for $S_n$, but I am not sure how to define such a basis).

Being a physicist, it is hard for me to understand if this is a simple problem, or an unsolvable one, so any advice is appreciated.