## 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_{2}n_{-}_{1} on the multilinear component of the free LAnKe with 2n - 1 generators is given by an irreducible representation whose dimension is the nth 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_{3}n_{-}_{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 ?, where ? has strictly decreasing column lengths, as a consequence of our eigenspace result.

Original language | English (US) |
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State | Published - 2018 |

Event | 30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018 - Hanover, United States Duration: Jul 16 2018 → Jul 20 2018 |

### Conference

Conference | 30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018 |
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Country/Territory | United States |

City | Hanover |

Period | 7/16/18 → 7/20/18 |

## Keywords

- Catalan numbers
- Free Lie algebra
- Specht modules

## ASJC Scopus subject areas

- Algebra and Number Theory