On a generalization of Lie($k$): a CataLAnKe theorem

TL;DR

This paper explores a new algebraic structure called free LAnKe, generalizing Lie algebras, and reveals its symmetric group representations relate to Catalan numbers, with implications for Specht modules.

Contribution

It introduces free LAnKe as an n-ary generalization of Lie algebras and characterizes the symmetric group representation on its multilinear component.

Findings

Representation dimension equals the nth Catalan number.

Provides a new presentation of staircase-shaped Specht modules.

Establishes a connection between free LAnKe and symmetric group eigenspaces.

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, which has additional consequences. We also obtain a new presentation of Specht modules of staircase shape as a consequence of our central result.