On the action of the symmetric group on the free LAnKe: a question of Friedmann, Hanlon, Stanley and Wachs
Mihalis Maliakas, Dimitra-Dionysia Stergiopoulou
TL;DR
The paper resolves a long-standing question about the symmetric-group action on the multilinear component $\mathrm{Lie}_n(m)$ of free LAnKe for all $n\ge k$ by linking the representation to skew Weyl modules and skew Specht modules. It constructs two maps $\Phi_1$ and $\Phi_2$ to control the Weyl-module presentation and shows that $\mathrm{Lie}_n(m)$ is a quotient of a skew Specht module $S^{\alpha(n,k)'}$, with $\alpha(n,k)$ chosen so that Littlewood–Richardson rules force vanishing of unwanted column-lengths when $n\ge k$. Through the duality functor $\Omega$ and the Schur functor $f$, this yields a surjection from $S^{\alpha(n,k)'}$ onto $\mathrm{Lie}_n(m)$ and proves $\gamma_{n,k}=0$ for all $n\ge k$, thereby giving the desired irreducible decomposition constraints. The approach also yields an explicit decomposition for $\rho_{n,4}$ (for $n\ge3$) and furnishes an upper bound on multiplicities in terms of Littlewood–Richardson coefficients, highlighting the representation-theoretic structure of the action. Overall, the results advance understanding of how the symmetric group acts on free $n$-ary Lie algebras and connect combinatorial representation theory with $GL_N$-module theory in a novel way.
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.