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

Abstract: 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.