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 $N\_d(\lambda,\mu)$? 



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


