TITLE: What is known about this plethysm?
QUESTION [31 upvotes]: Let $S^{\lambda}$ be a Schur functor. Is there a known positive rule to compute the decomposition of $S^{\lambda}(\bigwedge^2 \mathbb{C}^n)$ into $GL_n(\mathbb{C})$ irreps?

In response to Vladimir's request for clarification, the ideal answer would be a finite set whose cardinality is the multiplicity of $S^{\mu}(\mathbb{C}^n)$ in $S^{\lambda}(\bigwedge^2 \mathbb{C}^2)$. As an example, the paper Splitting the square of a Schur function into its symmetric and anti-symmetric parts gives such a rule for $\bigwedge^2(S^{\lambda}(\mathbb{C}^n))$.
Formulas involving evaluations of symmetric group characters, or involving alternating sums over stable rim hooks, are not good because they are not positive.
And, yes, it is easy to relate the answers for $\bigwedge^2 \mathbb{C}^n$ and $\mathrm{Sym}^2(\mathbb{C}^n)$, so feel free to answer with whichever is more convenient.

REPLY [7 votes]: Let $V = \mathbb{C}^n$ where $n$ is sufficiently large. This paper by Melanie de Boeck and Rowena Paget determines the constituents of $S^\lambda (\mathrm{Sym}^2 V)$ when $\lambda$ has either two rows, or two columns or is a hook partition of the form $(k-r,1^r)$. Since $S^\mu (V)$ appears in $S^\lambda (\mathrm{Sym}^2 V)$ if and only if $S^{\mu'}$ appears in $S^\lambda (\bigwedge^2V)$, these results apply to the question.
Explicit positive formulae are given for the multiplicities of irreducible consituents of $S^{(k-1,1)}(\mathrm{Sym}^2 V)$, $S^{(2,1^{k-2})}(\mathrm{Sym}^2V)$, $S^{(k-2,2)}(\mathrm{Sym}^2 V)$ and $S^{(k-2,1^2)}(\mathrm{Sym}^2 V)$. These results give a complete answer to the question in three new cases.
For example, Corollary 3.2 states that if $\mu$ is a partition of $2k$ then $S^\mu V$ appears in $S^{(k-1,1)}(\mathrm{Sym}^2 V)$ if and only if either $\mu$ has  only even parts, or $\mu$ has exactly two odd parts of distinct sizes. In the latter case the multiplicity is $1$, in the former case the multiplicity is one less than the number of distinct part sizes of $\mu$.
Edit. Say that $S^\lambda(V)$ is a minimal constituent of a polynomial $\mathrm{GL}(V)$-module $W$ if $S^\lambda(V)$ appears in $W$ and $\lambda$ is minimal with this property. Define maximal constituent analogously. Let $m \in \mathbb{N}$. This paper by Rowena Paget and me characterizes, in terms of certain tuples of families of $m$-subsets of $\mathbb{N}$, all partitions $\mu$ such that $S^\mu$ is a minimal  constituent of $S^\lambda(\mathrm{Sym}^m(V))$. 
There is an analogous characterization of the maximal constituents of $S^\lambda(\mathrm{Sym}^m(V))$ by replacing sets with multisets.
To give a very small example, the minimal constituent $S^{(4,3,1)}(V)$ of $S^{(1^4)}({\mathrm{Sym}^2(V)})$ corresponds to the family of $2$-sets $\bigl\{ \{1,2\}, \{1,3\}, \{2,3\}, \{1,4\} \bigr\}$ of multidegree $(4,3,1)' = (3,2,2,1)$.
These results give a practical sufficient condition on a partition $\nu$ for $S^\nu(V)$ to have multiplicity zero in $S^\lambda(\mathrm{Sym}^2(V))$, so are also relevant to the question.