Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module

Question

Let $V$ be a finite-dimensional, complex vector space and set $\newcommand{\Gl}{\mathrm{Gl}}G:=\Gl(V)\times\Gl(V)$. Let $E:=\mathrm{End}(V)$ and consider its coordinate ring $\mathbb C[E]$, the space of all polynomial functions on $E$. It is well-known (see 9.7 in Claudio Procesi's book on Lie Groups) that as a $G$-module, $\mathbb C[E]$ decomposes as $$\begin{align*} \mathbb C[E]_d &= \bigoplus_{\lambda\mathrel\vdash d} \mathbb S_\lambda(V^\ast)\otimes\mathbb S_\lambda(V) \end{align*}$$ where $\mathbb S_\lambda$ denotes the Schur functor. Now, simply by choosing a basis and restricting to permutation matrices, we have an action of $S:=S_n\times S_n$ on $E$ and therefore, also on $\mathbb C[E]$. Hence, there must be some decomposition $$\begin{align*} \mathbb C[E]_d &= \bigoplus_{\lambda,\mu \mathrel\vdash n} (\mathbb V_\lambda\otimes \mathbb V_\mu)^{\oplus N_d(\lambda,\mu)} \end{align*}$$ where $\mathbb V_\lambda$ is the Specht module and $N_d(\lambda,\mu)\in\mathbb N$ are certain multiplicities.

My question is: What, if anything, is known about the $N_d(\lambda,\mu)$?

Answer 1

You can give a formula for these numbers in terms of plethysm and internal product and inner product.

The starting point is Exercise 7.74 of Enumerative Combinatorics II by Richard Stanley which is a formula for the Schur functors of the defining representation of the symmetric group in terms of plethysm and inner product.