The representation $\text{Sym}(\text{Sym}^3(V))$ of $\text{GL}(V)$ decomposes into a direct sum of $S\_{\lambda}(V)$, where the $S\_{\lambda}$ are Schur functors. What is know about this decomposition?


I am aware that there isn't a complete solution currently, but what is known about $\text{Sym}^k(\text{Sym}^3(V))$ for various $k$?



What is known about the decomposition of $Sym(Sym^3(V))$ into irreducibles?
If $\lambda$ has at most two rows then the multiplicity of $S\_\lambda$ is given by the Cayley–Sylvester formula. This gives the complete decomposition if $\dim V = 2$. If $\lambda$ is of `near-hook' shape then there are relevant results in [this paper](http://arxiv.org/pdf/1207.6300v1.pdf) by Giannelli.


The unique maximal constituent is $S\_{(3k)}$. For any $m$, the minimal partitions $\lambda$ in the dominance order, such that $S\_\lambda$ appears in $\mathrm{Sym}^k(\mathrm{Sym}^m(V))$ are classified in a [joint paper](http://arxiv.org/pdf/1007.2946v2.pdf) with Rowena Paget. (For $m$ odd we later learned that this result follows from earlier [work of Klivans and Reiner](http://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r14).) To give a flavour of these results, when $k=m=3$, the minimal partitions are $(5,2,2)$ and $(4,4,1)$; the first corresponds to the set family $\{ \{1,2,3\},\{1,2,4\},\{1,2,5\} \}$ and the second to the set family $\{ \{1,2,3\}, \{1,2,4\}, \{1,3,4\} \}$. In the Klivans–Reiner setting, these become highest weight vectors in $\bigwedge^3\bigl(\bigwedge^3 V\bigr)$. For instance, if $V = \langle e\_1, e\_2, \ldots \rangle$ then the first corresponds to


$$(e\_1 \wedge e\_2 \wedge e\_3) \wedge (e\_1 \wedge e\_2 \wedge e\_4) \wedge (e\_1 \wedge e\_2 \wedge e\_5) \in \bigwedge^3\Bigl( \bigwedge^3V \Bigr).$$


Another result that can be proved using highest weight vectors is
Proposition 4.3.4 in [Ikenmeyer's thesis](http://math-www.uni-paderborn.de/agpb/work/ikenmeyer_thesis.pdf): if $S\_\lambda$ appears in $\mathrm{Sym}^k(\mathrm{Sym}^3(V))$ and $S\_\mu$ appears in $\mathrm{Sym}^\ell(\mathrm{Sym}^3(V))$ then $S\_{\lambda + \mu}$ appears in $\mathrm{Sym}^{k+\ell}(\mathrm{Sym}^3(V))$.


The $S\_\lambda$ appearing in $\mathrm{Sym}^3(\mathrm{Sym}^k(V))$ were found for all $k$ by Dent and Siemons, who showed they all appear, with at least the relevant multiplicity, in $\mathrm{Sym}^k(\mathrm{Sym}^3(V))$. This proves a special case of Foulkes' Conjecture.


For small $k$ the complete decomposition can be found using symmetric functions or other methods. There are [data](http://www.ma.rhul.ac.uk/~uvah099/other.html#Foulkes) on my website for $k \le 15$, computed using Proposition 5.1 in this [joint paper](http://arxiv.org/pdf/1202.0067v3.pdf) with Anton Evseev and Rowena Paget.


None of these results comes close to giving a complete picture. I have looked, without success, for a combinatorial rule that gives the constituents of $\mathrm{Sym}^{k+1}(\mathrm{Sym}^3(V))$ from $\mathrm{Sym}^{k}(\mathrm{Sym}^3(V))$, analogous to Young's Rule or Pieri's Rule.


