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Congruences of shifted Jack Littlewood-Richardson coefficients

T0 review · 0 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The shifted Jack Littlewood-Richardson coefficients for triples differing by a single box move are congruent modulo the α-hook length of the pivot box.

desk verdict This note proves the author's earlier conjecture on a modular congruence for shifted Jack LR coefficients when partitions differ by one box move. read the letter →

arxiv 2606.17822 v1 pith:SAIYSRPK submitted 2026-06-16 math.CO math.RT

Ryan Mickler This is my paper · ORCID
classification math.COmath.RT
keywords shiftedJackLittlewood-Richardsoncoefficientscongruencesboxmovesalpha-hooklengthsparameterpartitionsLaurentpolynomialssymmetricfunctions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a conjecture stating that the shifted Jack Littlewood-Richardson coefficients g^λ_μν(α) for two triples of partitions, in which one partition differs from the other by moving a single box, are congruent modulo the α-hook length of the pivot box for that move. These coefficients are Laurent polynomials in the Jack parameter α attached to triples of partitions and generalize the classical Jack Littlewood-Richardson coefficients. A sympathetic reader would care because the congruence supplies a direct relation between values of these polynomials at different partitions, which may streamline their study or computation. The proof settles the Jack case while the analogous statement for shifted Macdonald functions is left open.

What carries the argument

The shifted Jack Littlewood-Richardson coefficients g^λ_μν(α), Laurent polynomials in α attached to triples of partitions.

What would settle it

A concrete counterexample would be any specific triple of partitions and box move for which the two coefficients differ by a nonzero multiple of the α-hook length when evaluated at some value of α.

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Extended reading notes

Core claim

The central claim is that if two triples of partitions differ by a single box move in one of the partitions, then their associated shifted Jack Littlewood-Richardson coefficients are congruent modulo the α-hook length of the pivot box for that move.

Load-bearing premise

The shifted Jack Littlewood-Richardson coefficients are Laurent polynomials in the Jack parameter α.

Editorial extensions

If this is right

  • The stated congruence holds for every single-box move between valid triples.
  • The result applies uniformly for all values of the Jack parameter α at which the coefficients are defined.
  • The Macdonald-function analogue of the congruence remains unresolved because two required properties of Lassalle's shift map are not yet established.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The congruence might support recursive algorithms that compute the coefficients by reducing partition size one box at a time.
  • Specializing the parameter α to particular numbers could produce new numerical identities among ordinary Littlewood-Richardson coefficients.
  • The modular relation may connect to combinatorial interpretations or positivity properties that have not yet been examined.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript proves the author's earlier conjecture that the shifted Jack Littlewood-Richardson coefficients g^λ_μν(α) for two triples of partitions differing by a single box move are congruent modulo the α-hook length of the pivot box. The coefficients are Laurent polynomials in the Jack parameter α (as established by Alexandersson-Féray). The note also records that the analogous statement for shifted Macdonald functions remains open, pending two properties of Lassalle's shift map.

Significance. If the proof holds, the result supplies a concrete congruence relation for these generalized coefficients, extending classical Littlewood-Richardson theory to the Jack setting. The manuscript explicitly credits the prior definition of g^λ_μν(α) and isolates the precise obstruction for the Macdonald case, which is a useful clarification.

minor comments (2)
  1. [Abstract] The abstract refers to 'a previous work of the author's' for the conjecture; adding the precise citation (even if it is the author's own arXiv preprint) would improve traceability.
  2. Notation for the α-hook length and the 'pivot box' is used without an inline definition or forward reference to the section where it is introduced; a brief parenthetical reminder would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. The report contains no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; proof of prior conjecture is self-contained

full rationale

The manuscript states a conjecture from the author's prior work and then proves it in the present note. The background fact that the coefficients are Laurent polynomials in α is cited from Alexandersson-Féray (external authors) and is used only to make the modulo statement well-defined; it is not derived from the present result. No equations, fitted parameters, ansatzes, or uniqueness theorems are shown to reduce the main congruence to the inputs by construction. The derivation chain consists of a mathematical proof whose steps are independent of the conjecture statement itself. This is the normal case of a paper proving an earlier claim rather than circularly re-deriving its own premises.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the prior definition of the coefficients as Laurent polynomials and on the conjecture from the author's previous work; no free parameters or invented entities are introduced in the abstract.

assumptions (1)
  • domain assumption The shifted Jack Littlewood-Richardson coefficients are Laurent polynomials in the Jack parameter α
    Stated in the opening sentence of the abstract as the object of study

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Cite this review

Pith. "Pith review of Congruences of shifted Jack Littlewood-Richardson coefficients." pith.science (2026). https://pith.science/paper/SAIYSRPK

@misc{pith2026260617822,
  author       = {Pith},
  title        = {Pith review of: Congruences of shifted Jack Littlewood-Richardson coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SAIYSRPK}},
  note         = {Machine review of arXiv:2606.17822}
}
abstract

The shifted Jack Littlewood-Richardson coefficients $g^\lambda_{\mu\nu}(\alpha)$, first studied by Alexandersson-F\'eray, are Laurent polynomials in the Jack parameter $\alpha$ attached to triples of partitions, which generalize the classical Jack Littlewood-Richardson coefficients investigated by Stanley, et al. In a previous work of the author's, it was conjectured that the Littlewood-Richardson coefficients for two triples, in which one of the partitions differ by a single box move, are congruent modulo the $\alpha$-hook length of the pivot box for that move. In this note we prove that conjecture. We also investigate the extension of that conjecture to shifted Macdonald functions, which remains open pending two properies of Lassalle's shift map in that case.

Figures

Figures reproduced from arXiv: 2606.17822 by the authors.

Figure 1
Figure 1. For any partition, the possible pivot boxes p are given by the set of (non outer corner) boxes that either share a row with an addable box and a column with a removable box (red), or share a column with an addable box and a row with a removable box (blue). 1.2. Shifted coordinate congruence. The shifted variables zi = αxi − i are evaluated at a partition λ = (λ1, λ2, . . .) as zi(λ) = αλi − i. This present work begi… view at source ↗
Figure 2
Figure 2. The graded symmetric function Φµν = sh−1 (J # µ J # ν ) has top degree |µ| + |ν|, where its leading slice is the ordinary product JµJν. By Knop–Sahi vanishing Φ (k) µν = 0 for k < max(|µ|, |ν|), so its support is the band max(|µ|, |ν|) ≤ k ≤ |µ| + |ν|. 2.3. Output pivots. Proposition 2. Let λ ∼p λ˜ be a pivot with shared hook h. Then g λ µν ≡ g λ˜ µν (mod h). Proof. Recall from (5) and its sequel that J # µ J # ν = … view at source ↗
Figure 3
Figure 3. The function Ψλ/µ is supported in degrees |λ| − |µ| through |λ|. Its bottom slice is the ordinary skew Jack Jλ/µ, while its top slice is the single term J # µ (λ) Jλ, whose coefficient is proportional to the generalized Jack binomial coefficient λ µ  α = J # µ (λ)/J# µ (µ) of [8, 5]. 3. Extension to Macdonald functions We end by recording the parallel picture for shifted Macdonald polynomials. We will see that the … view at source ↗

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Works this paper leans on

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