## 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.

This paper is available as:

*Return to preprints page*