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arXiv · 1710.00376

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

Tamar Friedmann·Phil Hanlon·Richard P. Stanley·Michelle L. Wachs
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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.

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Tamar Friedmann, Phil Hanlon, Richard P. Stanley, Michelle L. Wachs. 2017-10-01. On a generalization of Lie($k$): a CataLAnKe theorem. https://doi.org/10.1016/j.aim.2021.107570

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