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arxiv: 2509.18625 · v2 · submitted 2025-09-23 · 🧮 math.CO

A formula for the Jack super nabla operator

Houcine Ben Dali This is my paper

Pith reviewed 2026-05-18 14:54 UTC · model grok-4.3

classification 🧮 math.CO
keywords Jack polynomialssuper nabla operatorpower-sum basisChapuy-Dołęga operatorsNazarov-Sklyanin operatorsHeisenberg algebrastructure coefficients
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The pith

The Jack super nabla operator admits a differential expression in the power-sum basis via Chapuy-Dołęga and Nazarov-Sklyanin operators.

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

The paper proves an explicit differential formula for a Jack polynomial version of the super nabla operator, written ∇(p, q). The formula expresses the operator's action on power sums using Chapuy-Dołęga operators combined with a dehomogenized form of Nazarov-Sklyanin operators. This expression is derived by first constructing a more general operator G(p, q) that records the structure coefficients of Jack characters and then extracting its highest-degree homogeneous component. A reader interested in symmetric functions would care because the result supplies a concrete computational handle on an operator that governs many algebraic and combinatorial properties of Jack polynomials.

Core claim

We prove that ∇(p, q) has a differential expression in the power-sum basis given in terms of Chapuy-Dołęga and Nazarov-Sklyanin operators. This result is obtained from a more general formula for the operator G(p, q) encoding the structure coefficients of Jack characters, from which ∇(p, q) is obtained by taking the top homogeneous part. A key step of the proof involves establishing that Chapuy-Dołęga operators together with a dehomogenized version of Nazarov-Sklyanin operators have a Heisenberg algebra structure. The proof also uses a characterization of the operator G(p, q) with a family of differential equations.

What carries the argument

The operator G(p, q) that encodes the structure coefficients of Jack characters; its top homogeneous part yields ∇(p, q).

If this is right

  • The Heisenberg algebra relation supplies the algebraic relations needed to simplify the differential expression for ∇(p, q).
  • Any identity previously known for the general operator G immediately specializes to an identity for its top part ∇(p, q).
  • The formula places the Jack super nabla on the same computational footing as existing operators already used for ordinary Jack characters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Heisenberg-algebra technique might produce analogous differential formulas for other deformed nabla-type operators in the Jack setting.
  • Explicit expressions of this kind could be used to test conjectural positivity or integrality properties of coefficients that appear when the super nabla acts on Jack polynomials.

Load-bearing premise

The operator G(p, q) is fully characterized by a family of differential equations whose top-degree solution is the desired nabla operator.

What would settle it

Directly apply the proposed differential expression to a low-degree power-sum symmetric function and check whether the result equals the known action of the Jack super nabla operator on the corresponding Jack character.

read the original abstract

We study a Jack analog $\nabla(\mathbf{p},\mathbf{q})$ of the super nabla operator recently introduced by Bergeron, Haglund, Iraci and Romero for Macdonald polynomials. We prove that $\nabla(\mathbf{p},\mathbf{q})$ has a differential expression in the power-sum basis given in terms of Chapuy-Do\l{}e\k{}ga and Nazarov-Sklyanin operators. This result is obtained from a more general formula for the operator $G(\mathbf{p},\mathbf{q})$ encoding the structure coefficients of Jack characters, from which $\nabla(\mathbf{p},\mathbf{q})$ is obtained by taking the top homogeneous part. A key step of the proof involves establishing that Chapuy-Do\l{}e\k{}ga operators together with a dehomogenized version of Nazarov-Sklyanin operators have a Heisenberg algebra structure. The proof also uses a characterization of the operator $G(\mathbf{p},\mathbf{q})$ with a family of differential equations, recently established by the author.

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.

Referee Report

1 major / 2 minor

Summary. The paper proves that the Jack super nabla operator ∇(p,q) admits a differential expression in the power-sum basis expressed using Chapuy-Dołęga and dehomogenized Nazarov-Sklyanin operators. This is derived from a general formula for the structure-coefficient operator G(p,q) by extracting its top homogeneous part, utilizing Heisenberg algebra relations among the operators and a prior characterization of G via differential equations established by the author.

Significance. If valid, this result provides an explicit algebraic formula for the Jack analog of the super nabla operator, building on work for Macdonald polynomials. It strengthens the connection between combinatorial operators and algebraic structures in symmetric function theory. The proof of the Heisenberg relations is a positive technical contribution that could be useful in related contexts.

major comments (1)
  1. The derivation of the top homogeneous part for ∇(p,q) from G(p,q) depends on the family of differential equations characterizing G(p,q) from the author's recent prior work. The manuscript cites this characterization but does not reprove or adapt it specifically to the super Jack setting. If the differential equations do not precisely capture the top-degree terms under the super deformation, the claimed formula does not follow. A section clarifying the applicability or providing an independent derivation of the relevant parts would address this.
minor comments (2)
  1. Ensure that references to the prior work on G(p,q) are clearly distinguished from the new contributions in this manuscript.
  2. The use of bold p and q for vectors of variables is standard but confirm consistency throughout the power-sum basis expressions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the significance of our results and for the constructive major comment. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The derivation of the top homogeneous part for ∇(p,q) from G(p,q) depends on the family of differential equations characterizing G(p,q) from the author's recent prior work. The manuscript cites this characterization but does not reprove or adapt it specifically to the super Jack setting. If the differential equations do not precisely capture the top-degree terms under the super deformation, the claimed formula does not follow. A section clarifying the applicability or providing an independent derivation of the relevant parts would address this.

    Authors: We thank the referee for highlighting this aspect of the proof structure. The family of differential equations characterizing G(p,q) was derived in our prior work in a general algebraic setting whose hypotheses are satisfied verbatim by the super Jack parameters p and q; the super deformation is built into the definition of the operators from the beginning, and the homogeneity arguments used to extract the top-degree part for ∇(p,q) carry over without modification. Nevertheless, to make the applicability fully explicit, we will insert a short clarifying subsection (approximately one paragraph) that recalls the relevant statements from the prior characterization, notes that the super Jack case falls inside the established framework, and confirms that the top-homogeneous extraction remains valid under the super grading. This addition addresses the concern without requiring a complete independent re-derivation of the differential equations. revision: yes

Circularity Check

1 steps flagged

Derivation of top homogeneous part for ∇(p,q) depends on prior differential-equation characterization of G(p,q) by the same author

specific steps
  1. self citation load bearing [Abstract]
    "This result is obtained from a more general formula for the operator G(p,q) encoding the structure coefficients of Jack characters, from which ∇(p,q) is obtained by taking the top homogeneous part. ... The proof also uses a characterization of the operator G(p,q) with a family of differential equations, recently established by the author."

    The formula for ∇(p,q) is obtained by taking the top homogeneous part of G(p,q), whose characterization via differential equations is invoked directly from the author's own recent prior paper. The present work does not reprove this characterization in the super Jack context, so the final differential expression for ∇ depends on that self-cited result for isolating the precise top-degree terms.

full rationale

The paper derives the claimed differential expression for ∇(p,q) by extracting the top homogeneous component from a general formula for the structure-coefficient operator G(p,q). A key supporting step is the use of a family of differential equations that characterize G(p,q), which is explicitly cited as recently established by the same author in prior work. While the manuscript adds independent content by proving a Heisenberg algebra structure for Chapuy-Dołęga operators combined with a dehomogenized Nazarov-Sklyanin operator, the load-bearing characterization of G itself is not reproved or independently verified inside the super setting. This creates moderate dependence on self-citation without reducing the entire derivation to a tautology or fit. The central algebraic steps remain non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on standard algebraic properties of symmetric-function operators and a prior self-established characterization of G; no free parameters or new postulated entities appear.

axioms (2)
  • domain assumption Chapuy-Dołęga operators together with dehomogenized Nazarov-Sklyanin operators satisfy Heisenberg algebra commutation relations
    Invoked as the key technical step to obtain the differential expression.
  • ad hoc to paper The operator G(p,q) is characterized by a family of differential equations
    Established by the author in previous work and used to derive the top homogeneous part for ∇.

pith-pipeline@v0.9.0 · 5698 in / 1461 out tokens · 54665 ms · 2026-05-18T14:54:57.284707+00:00 · methodology

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