On the action of the symmetric group on the free LAnKe: a question of Friedmann, Hanlon, Stanley and Wachs

TL;DR

This paper investigates the representation of the symmetric group acting on free LAnKe algebras, confirming a conjecture about the structure of its irreducible components for all values of k.

Contribution

It proves that the irreducible decomposition contains no Young diagram with at most k-1 columns for all k, extending previous results limited to k ≤ 3.

Findings

Confirmed the conjecture for all k

Provided a new proof different from previous work

Extended understanding of symmetric group actions on free LAnKe

Abstract

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.