## Action of the symmetric group on the free LAnKe: a CataLAnKe Theorem

Tamar Friedmann, Philip Hanlon, Richard P. Stanley, Michelle L. Wachs

Abstract: We initiate a study of the representation of the symmetric group on the multilinear component of an $n$-ary generalization of the free Lie algebra, which we call a free LAnKe. Our central result is that the representation of the symmetric group $S_{2n-1}$ on the multilinear component of the free LAnKe with $2n-1$ generators is given by an irreducible representation whose dimension is the $n$th Catalan number. This leads to a more general result on eigenspaces of a certain linear operator. A decomposition, into irreducibles, of the representation of $S_{3n-2}$ on the multilinear component the free LAnKe with $3n-2$ generators is also presented. We also obtain a new presentation of Specht modules of shape $\lambda$, where $\lambda$ has strictly decreasing column lengths, as a consequence of our eigenspace result.

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