On an $n$-ary generalization of the Lie representation and tree Specht modules

We continue our study, initiated in our prior work with Richard Stanley, of the representation of the symmetric group on the multilinear component of an $n$-ary generalization of the free Lie algebra known as the free Filippov $n$-algebra with $k$ brackets. Our ultimate aim is to determine the multiplicities of the irreducible representations in this representation. This had been done for the ordinary Lie representation ($n=2$ case) by Kraskiewicz and Weyman. The $k=2$ case was handled in our prior work, where the representation was shown to be isomorphic to $S^{2^{n-1}1}$. In this paper, for general $n$ and $k$, we obtain decomposition results that enable us to determine the multiplicities in the $k=3$ and $k=4$ cases. In particular we prove that in the $k=3$ case, the representation is isomorphic to $S^{3^{n-1}1}\oplus S^{3^{n-2}{21}^2}$. Our main result shows that the multiplicities stabilize in a certain sense when $n$ exceeds $k$. As an important tool in proving this, we present two types of generalizations of the notion of Specht module that involve trees.