Q: Effect of Schur functors on Specht modules. Let $\lambda$ be a partition of $n$, $\mu$ a partition of $m$, $S_\lambda$ be the Schur functor associated to $\lambda$ and $V^\mu$ the Specht module associated to $\mu$, i.e. the irreducible representation of the symmetric group $\Sigma_m$ in $m$ letters corresponding to the partition $\mu$.
By functoriality, $S_\lambda(V^\mu)$ is a new representation of $\Sigma_m$, so we can write it as a sum $$S_\lambda(V^\mu)=\bigoplus\limits_{\vartheta}c_{\lambda,\mu}^\vartheta V^{\vartheta}$$ of irreducibles.
I am interested in the coefficients $c_{\lambda,\mu}^\vartheta$. I skimmed some textbooks on representation theory, but could only find discussions about the effect of Schur functors on $GL_n$-representations instead of the ones of the symmetric groups.

A: There isn't a whole lot in the literature about these coefficients, they are closely related to so called plethysm coefficients which give the decomposition of the composition of two Schur functors into Schur functors.
In particular, to my knowledge there are no known combinatorial objects that these numbers count, and in fact such a combinatorial interpretation would probably be very interesting to people working in combinatorial representation theory.
I'll note that even the special case of applying a Schur functor to the standard $m$-dimensional representation of $\Sigma_m$ is not fully understood and it amounts to computing the restriction of irreducible $GL_m(\mathbb{C})$ representations to the copy of $\Sigma_m$ siting inside it as permutation matrices.
