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 S2n-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 S3n-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 | 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