Let $V$ be the (complex) vector space generated by words of length $nm$ where each letter from $1$ to $m$ appears exactly $n$ times. For example, if $m=2$ and $n=3$, then
$$ V = \mathbb{C}\{111222, 112122, 112212, \ldots\} $$
(In this example, V is of dimension $\binom{6}{3} = 20$).
This vector space has an action $\cdot$ on the symmetric group $\mathfrak{S}_{nm}$ that permutes the order of the letters in the word, and an action $\star$ of $\mathfrak{S}_{m}$ which permutes the values of the letters. For example,
$$ (134) \cdot 111222 = 211122 \ \ \text{ and } \ \ (12) \star 111222 = 222111 $$
It is easy to see that these actions commute, so $V$ is a representation of $\mathfrak{S}_{nm} \times \mathfrak{S}_{m}$.
It is known that the irreducible representations of this group are of the form $S^{\lambda} \otimes S^{\mu}$, where $\lambda$ is a partition of $nm$, $\mu$ is a partition of $m$, and each component of the tensor product is an irreducible representation of the symmetric group.
My question is : Is the decomposition of $V$ in irreducible representations of $\mathfrak{S}_{nm} \times \mathfrak{S}_{m}$ known?
As a representation of $\mathfrak{S}_{nm}$, it is isomorphic to the permutation module $M^{\overbrace{(n,\ldots, n)}^{m \text{ times}}}$, which can be thought of as the trivial representation of $\overbrace{\mathfrak{S}_{n} \times \ldots \times \mathfrak{S}_{n}}^{m \text{ times}}$ induced to $\mathfrak{S}_{nm}$.
As a representation of $\mathfrak{S}_{m}$, each orbit of a word yields to a copy of the regular representation, because each element of an orbit can be identified with a permutation by choosing the word corresponding to the identity.
I know how to decompose each of these representations into irreducible ones.
Knowing this, how can I understand the decomposition of $V$ into irreducible representations of $\mathfrak{S}_{nm} \times \mathfrak{S}_{m}$? It seems like I only miss one step, but I cannot find what it is…
