I am looking for a proof about the link between plethysm and wreath product. It is a well-known fact, being use extensively in many papers, but I can't find a good reference. Everything that follows can be found in Stanley's enumerative combinatorics, Volume 2, except for the proof I am looking for (the result is stated without proof at the last text page).
Let $U$ be a representation of the symmetric group $\mathfrak{S}_n$. It is known that such a representation corresponds to a symmetric function $f$ of degree $n$, via a map known as the Frobenius characteristic map (which involves the character of $U$).
Now let $V$ be a representation of $\mathfrak{S}_m$, and let $g$ be the corresponding symmetric function (of degree $m$). There is an operation on symmetric functions, called plethysm and denoted $f[g]$, which essentially works as a monomial substitution, meaning that it is the symmetric function obtained by replacing each variable of $f$ by a (monic) monomial appearing in $g$. In perticular, the degree of $f[g]$ is $nm$.
We now want a representation of $\mathfrak{S}_{nm}$ such that the Frobenius characteristic map sends this representation to $f[g]$. To do so, we first consider the wreath product $\mathfrak{S}_m \wr \mathfrak{S}_n$, which can be thought of as the subgroup of $\mathfrak{S}_{nm}$ which stabilizes the set partition
$$\{\{1, \ldots, m\},\{m+1, \ldots, 2m\}, \ldots, \{(n-1)m+1, \ldots, nm\}\}.$$
More precisely, it is the semi-direct product $\mathfrak{S}_m^n \rtimes \mathfrak{S}_n$, where $\mathfrak{S}_n$ acts on $\mathfrak{S}_m^n$ by permuting the components (the above characterization as a stabilizer gives an embedding into $\mathfrak{S}_{nm}$). Thus, an element of $\mathfrak{S}_m \wr \mathfrak{S}_n$ can be written as $(\sigma_1, \ldots, \sigma_n ; \pi)$, where $\sigma_i \in \mathfrak{S}_m$ for all $i$ and $\pi \in \mathfrak{S}_n$.
We combine $U$ and $V$ into a representation of $\mathfrak{S}_m \wr \mathfrak{S}_n$, by considering the vector space $V^{\otimes n} \otimes U$, where $(\sigma_1, \ldots, \sigma_n ; \pi) \in \mathfrak{S}_m \wr \mathfrak{S}_n$ acts on $v_1 \otimes \ldots \otimes v_n \otimes u$ in the following way:
$$(\sigma_1, \ldots, \sigma_n ; \pi) \cdot (v_1 \otimes \ldots \otimes v_n \otimes u) = (\sigma_{1} \cdot v_{\pi^{-1}(1)}) \otimes \ldots \otimes (\sigma_{n} \cdot v_{\pi^{-1}(n)}) \otimes (\pi \cdot u).$$
Theorem : The Frobenius characteristic map sends $(V^{\otimes n} \otimes U) \uparrow_{\mathfrak{S}_m \wr \mathfrak{S}_n}^{\mathfrak{S}_{nm}}$ to $f[g]$.
As I said, this theorem appears without proof in Stanley's Enumerative Combinatorics Volume 2. It is also widely used in the litterature. I know enough about both plethysm and wreath product to understand why it is true, but I am looking for a reference with a proof to include it in a paper I am writing.