On the action of the symmetric group on the free LAnKe

A LAnKe (also known as a Filippov algebra or a Lie algebra of the \(n\)-th kind) is a vector space equipped with a skew-symmetric \(n\)-linear form that satisfies the generalized Jacobi identity. Friedmann, Hanlon, Stanley and Wachs have shown that the symmetric group acts on the multilinear compone...

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Published in:arXiv.org 2024-02
Main Authors: Maliakas, Mihalis, Stergiopoulou, Dimitra-Dionysia
Format: Article
Language:English
Subjects:
Group theory
Lie groups
Representations
Vector spaces
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Summary:A LAnKe (also known as a Filippov algebra or a Lie algebra of the \(n\)-th kind) is a vector space equipped with a skew-symmetric \(n\)-linear form that satisfies the generalized Jacobi identity. Friedmann, Hanlon, Stanley and Wachs have shown that the symmetric group acts on the multilinear component of the free LAnKe on \(2n-1\) generators as an irreducible representation. They have claimed that the multilinear component on \(3n-2\) generators decomposes as a direct sum of two irreducible symmetric group representations. This paper provides a proof of the later statement.
ISSN:2331-8422