For each positive integer $i$, let $A\_i$ be a fixed representation of the symmetric group $S\_i$. I won't tell you exactly what $A\_i$ is, but let's say that I have a very explicit description of its character.


Let $\lambda$ be a partition of a fixed integer $n$, and let $\ell$ be its length. Let $W\_\lambda \subset S\_n$ be the stabilizer of $\lambda$. This group $W\_\lambda$ is a semidirect product of the Young subgroup $S\_\lambda = S\_{\lambda\_1}\times\ldots\times S\_{\lambda\_\ell}$ and a subgroup $T\_\lambda \subset S\_\ell$, which shuffles pieces of $\lambda$ of the same size.


How can one compute the character of $\operatorname{Ind}\_{W\_\lambda}^{S\_n}(A\_{\lambda\_1}\otimes\ldots \otimes A\_{\lambda\_\ell})$? Here $S\_{\lambda\_i}\subset S\_\lambda$ acts on the $i^\text{th}$ factor of the tensor product, and $T\_\lambda$ acts by permuting the factors of the tensor product.


In theory, I know that I have all of the necessary information to compute this character, but I wonder if there are any useful combinatorial tricks to organize the data.


Regarding the motivation for this question, I would like to apply the recursive formula in Remark 3.6 of this paper <http://pages.uoregon.edu/njp/ukl-m1.pdf> when $W=S\_n$ and $\mathcal{A}$ is the braid arrangement.



Inducing representations from the stabilizer of a partition
Let $\mathrm{ch}(A\_i)$ denote the Frobenius characteristic of $A\_i$
(as defined e.g. in *Enumerative Combinatorics*, vol. 2, page
351). Thus $\mathrm{ch}(A\_i)$ is a symmetric function. If $\lambda$
has $m\_i$ parts equal to $i$, then the Frobenius characteristic of
your induced character is given by $\prod\_i
h\_{m\_i}[\mathrm{ch}(A\_i)]$, where the brackets denote plethysm. See
for instance page 449 of the previous reference for a statement of a
somewhat more general result. The proof is a fairly straightforward
computation. Since plethysm is notoriously difficult to work with,
this interpretation will probably not be of much use for your
question about computing the character.


