On the action of the symmetric group on the free LAnKe: a question of Friedmann, Hanlon, Stanley and Wachs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908273877188608 |
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| author | Maliakas, Mihalis Stergiopoulou, Dimitra-Dionysia |
| author_facet | Maliakas, Mihalis Stergiopoulou, Dimitra-Dionysia |
| contents | A LAnKe (also known as a Lie algebra of the $n$th kind, or a Filippov algebra) is a vector space equipped with a skew-symmetric $n$-linear form that satisfies the generalized Jacobi identity. The symmetric group $\mathfrak{S}_m$ acts on the multilinear part of the free LAnKe on $m=(n-1)k+1$ generators, where $k$ is the number of brackets, by permutation of the generators. The corresponding representation was studied by Friedmann, Hanlon, Stanley and Wachs, who asked whether for $n \ge k$, its irreducible decomposition contains no summand whose Young diagram has at most $k-1$ columns. The answer is affirmative if $k \le 3$. In this paper, we show that the answer is affirmative for all $k$. A proof has been given recently by Friedmann, Hanlon and Wachs. The two proofs are completely different. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06979 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the action of the symmetric group on the free LAnKe: a question of Friedmann, Hanlon, Stanley and Wachs Maliakas, Mihalis Stergiopoulou, Dimitra-Dionysia Representation Theory Combinatorics 20C30, 05E10, 20G05 A LAnKe (also known as a Lie algebra of the $n$th kind, or a Filippov algebra) is a vector space equipped with a skew-symmetric $n$-linear form that satisfies the generalized Jacobi identity. The symmetric group $\mathfrak{S}_m$ acts on the multilinear part of the free LAnKe on $m=(n-1)k+1$ generators, where $k$ is the number of brackets, by permutation of the generators. The corresponding representation was studied by Friedmann, Hanlon, Stanley and Wachs, who asked whether for $n \ge k$, its irreducible decomposition contains no summand whose Young diagram has at most $k-1$ columns. The answer is affirmative if $k \le 3$. In this paper, we show that the answer is affirmative for all $k$. A proof has been given recently by Friedmann, Hanlon and Wachs. The two proofs are completely different. |
| title | On the action of the symmetric group on the free LAnKe: a question of Friedmann, Hanlon, Stanley and Wachs |
| topic | Representation Theory Combinatorics 20C30, 05E10, 20G05 |
| url | https://arxiv.org/abs/2410.06979 |