On the free LAnKe on $3n-2$ generators: a theorem of Friedmann, Hanlon, Stanley and Wachs

Dimitra-Dionysia Stergiopoulou; Mihalis Maliakas

arxiv: 2401.09405 · v4 · pith:K4M3MWWOnew · submitted 2024-01-17 · 🧮 math.RT · math.CO

On the free LAnKe on 3n-2 generators: a theorem of Friedmann, Hanlon, Stanley and Wachs

Mihalis Maliakas , Dimitra-Dionysia Stergiopoulou This is my paper

Pith reviewed 2026-05-24 04:25 UTC · model grok-4.3

classification 🧮 math.RT math.CO

keywords LAnKeFilippov algebrafree algebramultilinear componentsymmetric groupirreducible representationsgeneralized Jacobi identityrepresentation theory

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The pith

The multilinear component of the free LAnKe on 3n-2 generators decomposes as the direct sum of two irreducible symmetric group representations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that for a LAnKe, a vector space equipped with a skew-symmetric n-linear bracket obeying the generalized Jacobi identity, the multilinear part of the free algebra generated by 3n-2 elements splits into exactly two irreducible representations of the symmetric group S_{3n-2}. This extends the known irreducible case for 2n-1 generators. The authors supply an alternative proof to the one announced by Friedmann, Hanlon, Stanley and Wachs. A reader cares because the result describes how permutations of the generators organize the structure of these higher-arity algebras in a simple way.

Core claim

The multilinear component on 3n-2 generators of the free LAnKe decomposes as a direct sum of two irreducible symmetric group representations. The proof proceeds by establishing that the space is spanned by two distinct irreducible modules under the natural action that permutes the generators while preserving the defining relations.

What carries the argument

The symmetric group action on the multilinear component induced by permuting the generators, which preserves the generalized Jacobi identity.

Load-bearing premise

The symmetric group action on the multilinear part is well-defined via the standard construction and remains compatible with the generalized Jacobi identity.

What would settle it

For n=2, compute the multilinear component explicitly on 4 generators and check whether its dimension and character equal the sum of the dimensions and characters of exactly two irreducible representations of S_4.

read the original abstract

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 part of the free LAnKe on $2n-1$ generators as an irreducible representation. They announced that the multilinear component on $3n-2$ generators decomposes as a direct sum of two irreducible symmetric group representations and a proof was given recently in a subsequent paper by Friedmann, Hanlon and Wachs. In the present paper we provide a proof of the later statement. The two proofs are substantially different.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is an alternative proof of a decomposition result already established by Friedmann et al.

read the letter

The main takeaway is that this paper supplies a different proof of the claim that the multilinear part of the free LAnKe on 3n-2 generators decomposes into two irreducible representations of S_{3n-2}. Friedmann, Hanlon, Stanley and Wachs announced the result and proved it in a later paper; the current work positions its argument as substantially distinct from that one. That is the extent of what is new here. The paper does the straightforward job of writing down the alternative derivation using the standard multilinear setup and the generalized Jacobi identity. If the steps check out, readers get a second route to the same decomposition, which can matter when the original proof relies on tools that do not generalize easily. The construction of the symmetric group action is the usual one, so the compatibility with the identity for 3n-2 generators follows from the same skew-symmetry and identity that work for 2n-1; the stress-test concern does not appear to create an extra obstacle. The soft spot is simply that the theorem is not new. The contribution therefore reduces to the proof technique itself. If that technique is shorter, more elementary, or suggests further calculations, the paper earns its place; otherwise its audience stays narrow. The paper is aimed at people already working on representation theory of n-ary algebras who want explicit decompositions or who might adapt the method. It is the sort of concrete, self-contained argument that deserves a serious referee even though the statement was known beforehand. I would send it to review.

Referee Report

0 major / 1 minor

Summary. The manuscript provides an alternative proof, substantially different from that of Friedmann-Hanlon-Wachs, that the multilinear component of the free LAnKe on 3n-2 generators decomposes as a direct sum of two irreducible representations of the symmetric group S_{3n-2}.

Significance. The result determines the S_{3n-2}-module structure of the multilinear part in this case, extending the known irreducibility for 2n-1 generators. The independent proof is a strength of the manuscript.

minor comments (1)

The introduction would benefit from a short paragraph outlining the main steps of the new proof and how it differs from the existing one.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, which accurately summarizes the contribution of our manuscript as an independent proof of the two-irrep decomposition. We are pleased by the recommendation to accept.

Circularity Check

0 steps flagged

Independent algebraic proof with no reduction to self-definition or fitted inputs

full rationale

The paper states it supplies a substantially different proof of the announced decomposition result for the multilinear component of the free LAnKe on 3n-2 generators. The symmetric-group action is invoked via the standard construction already used for the 2n-1 case, and the generalized Jacobi identity is treated as part of the given algebraic structure rather than derived from the target representation. No equations are shown to be tautological by construction, no parameters are fitted to data and then relabeled as predictions, and the cited prior work is by different authors. The derivation therefore remains self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard definition of LAnKe algebras and on basic facts from representation theory of the symmetric group; no new free parameters or entities are introduced.

axioms (2)

domain assumption A LAnKe is a vector space equipped with a skew-symmetric n-linear form satisfying the generalized Jacobi identity.
Core definition invoked throughout the abstract.
standard math The symmetric group acts on the multilinear component by permuting generators.
Standard construction in free algebra representation theory.

pith-pipeline@v0.9.0 · 5676 in / 1075 out tokens · 41009 ms · 2026-05-24T04:25:33.355175+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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Friedmann T., Hanlon P., Stanley R. P. and Wachs M. L., On a generalization of Lie(k): a CataLAnKe theorem, Adv. Math. 380 (2021), 107570

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and Wachs M

Friedmann T., Hanlon P. and Wachs M. L., A new presentation of Specht modules with distinct parts, Electron. J. Comb. 31(4) (2024), P4.41

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[1] [1]

and Buchsbaum, D., Characteristic- free representation theory of the general linear group II: Homological considerations, Adv

Akin K. and Buchsbaum, D., Characteristic- free representation theory of the general linear group II: Homological considerations, Adv. Math. 72 (1988), 172-210

work page 1988

[2] [2]

and Friedmann T., A simplified presentation of Specht modules, J

Brauner S. and Friedmann T., A simplified presentation of Specht modules, J. Pure Appl. Algebra 226 (2022), 106979

work page 2022

[3] [3]

de Azc´ arraga J. A. and Izquierdo J. M., n-ary algebras: a review with applications, J. Phys. A: Math. Theor. 43 (2010), 293001

work page 2010

[4] [4]

T., n-Lie algebras, Sib

Filippov V. T., n-Lie algebras, Sib. Mat. Zh. 26 (1985), 126-140. (English translation: Siberian Math. J. 26 (1985), no. 6, 879–891)

work page 1985

[5] [5]

Friedmann T., Hanlon P., Stanley R. P. and Wachs M. L., Action of the symmetric group on the free LAnKe: a CataLAnKe Theorem, Seminaire Lothoringien de Combinitoire 80B (2018), No. 63

work page 2018

[6] [6]

Friedmann T., Hanlon P., Stanley R. P. and Wachs M. L., On a generalization of Lie(k): a CataLAnKe theorem, Adv. Math. 380 (2021), 107570

work page 2021

[7] [7]

and Wachs M

Friedmann T., Hanlon P. and Wachs M. L., A new presentation of Specht modules with distinct parts, Electron. J. Comb. 31(4) (2024), P4.41

work page 2024

[8] [8]

and Wachs M

Friedmann T., Hanlon P. and Wachs M. L., On an n-ary generalization of the Lie representation and tree Specht modules, arXiv:2402.19174

work page arXiv

[9] [9]

35, Cambridge University Press, 1997

Fulton W., Young Tableaux, With Applications to Representation Theory and Geometry, vol. 35, Cambridge University Press, 1997

work page 1997

[10] [10]

A., Polynomial Representations of GLn, 2nd edition, LNM 830, Springer, 2007

Green J. A., Polynomial Representations of GLn, 2nd edition, LNM 830, Springer, 2007

work page 2007

[11] [11]

and Stergiopoulou D.-D., Presentations of Schur and Specht modules in characteristic zero, J

Maliakas M., Metzaki M. and Stergiopoulou D.-D., Presentations of Schur and Specht modules in characteristic zero, J. Pure Appl. Algebra 229 (2025), 107774

work page 2025

[12] [12]

and Stergiopoulou D.-D., On homomorphisms into Weyl modules corresponding to partitions with two parts, Glasgow Math

Maliakas M. and Stergiopoulou D.-D., On homomorphisms into Weyl modules corresponding to partitions with two parts, Glasgow Math. J. 65 (2023), 272–283

work page 2023

[13] [13]

and Stergiopoulou D.-D., On the action of the symmetric group on the free LAnKe II, arXiv:arXiv:2410.06979

Maliakas M. and Stergiopoulou D.-D., On the action of the symmetric group on the free LAnKe II, arXiv:arXiv:2410.06979

work page arXiv

[14] [14]

149 Cambridge University Press, 2003

Weyman J., Cohomology of vector bundles and syzygies, vol. 149 Cambridge University Press, 2003. Department of Mathematics, University of Athens, Greece Email address: mmaliak@math.uoa.gr Department of Mathematics, University of Athens, Greece Email address: dstergiop@math.uoa.gr 29

work page 2003