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arxiv: 2404.03904 · v2 · pith:65LLTH2Snew · submitted 2024-04-05 · 🧮 math.CO

Macdonald characters from a new formula for Macdonald polynomials

Houcine Ben Dali , Michele D'Adderio This is my paper

Pith reviewed 2026-05-24 01:51 UTC · model grok-4.3

classification 🧮 math.CO
keywords Macdonald polynomialssymmetric functionsJack charactersMacdonald characterspositivity conjecturescreation formulashifted Macdonald polynomials
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The pith

A new operator on symmetric functions produces a creation formula for Macdonald polynomials that links two prior theories.

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

The paper introduces an operator called Gamma on symmetric functions. This operator supplies a creation formula for Macdonald polynomials. The formula connects the approach based on Macdonald operators with the one based on shifted Macdonald polynomials. The authors then define Macdonald characters as a two-parameter generalization of Jack characters. They also give a change of variables that lets them state positivity conjectures extending known open problems for Jack polynomials.

Core claim

The central claim is that the operator Gamma yields an explicit creation formula for Macdonald polynomials, thereby establishing a direct link between the Macdonald operators theory and the shifted Macdonald polynomials theory, which in turn permits the definition of Macdonald characters and the formulation of associated positivity conjectures via a change of variables.

What carries the argument

The operator Gamma on symmetric functions, which generates the creation formula for Macdonald polynomials.

If this is right

  • Macdonald polynomials admit an explicit creation formula constructed from the action of Gamma.
  • Macdonald characters form a two-parameter family that generalizes Jack characters.
  • Positivity conjectures for Macdonald characters can be stated after the given change of variables.
  • Open problems on Jack polynomials, including those of Goulden and Jackson, extend directly to the Macdonald setting.

Where Pith is reading between the lines

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

  • The creation formula may supply new combinatorial rules for expanding Macdonald polynomials in other bases.
  • Computational checks of the positivity conjectures for small partitions could provide early evidence or counterexamples.
  • The link between the two theories might allow transfer of known results from one setting to the other.

Load-bearing premise

The operator Gamma is assumed to produce the stated creation formula that correctly connects the two theories of Macdonald polynomials.

What would settle it

Explicit computation of a low-degree Macdonald polynomial via the Gamma-derived creation formula that fails to match the known Macdonald polynomial would disprove the claim.

read the original abstract

We introduce a new operator $\Gamma$ on symmetric functions, which enables us to obtain a creation formula for Macdonald polynomials. This formula provides a connection between the theory of Macdonald operators initiated by Bergeron, Garsia, Haiman and Tesler, and shifted Macdonald polynomials introduced by Knop, Lassalle, Okounkov and Sahi. We use this formula to introduce a two-parameter generalization of Jack characters, which we call Macdonald characters. Finally, we provide a change of variables in order to formulate several positivity conjectures related to these generalized characters. Our conjectures extend some important open problems on Jack polynomials, including some famous conjectures of Goulden and Jackson.

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

0 major / 2 minor

Summary. The paper introduces a new operator Γ on symmetric functions that yields a creation formula for Macdonald polynomials. This is used to connect the Macdonald operators theory of Bergeron-Garsia-Haiman-Tesler with the shifted Macdonald polynomials of Knop-Lassalle-Okounkov-Sahi. The authors then define two-parameter Macdonald characters generalizing Jack characters and formulate several positivity conjectures extending those of Goulden-Jackson and related open problems on Jack polynomials.

Significance. If the operator Γ and the creation formula are correctly constructed without circularity or hidden parameters, the work would provide a concrete bridge between two established lines of research on Macdonald polynomials, potentially enabling new combinatorial interpretations and proofs of positivity. The Macdonald characters and the extended conjectures could open avenues for studying positivity in the two-parameter setting, building directly on known Jack polynomial problems.

minor comments (2)
  1. The abstract and title emphasize the new operator and formula, but without explicit verification examples or small-case computations in the provided overview, it is difficult to immediately confirm the formula reproduces known Macdonald polynomials.
  2. Notation for the new operator Γ and the Macdonald characters should be checked for consistency with standard references (e.g., Macdonald's book or the cited works) to avoid reader confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and accurate summary of the manuscript. The report correctly identifies the introduction of the operator Γ, the creation formula, the connection between the two lines of research, and the definition of Macdonald characters together with the extended positivity conjectures. No specific major comments appear under the MAJOR COMMENTS heading, so we offer no point-by-point replies at this time.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces an original operator Γ to derive a creation formula for Macdonald polynomials and defines Macdonald characters as a two-parameter generalization of Jack characters. No load-bearing step reduces by construction to its own inputs, fitted parameters, or self-citation chains; the derivation chain begins from the new operator definition and connects to prior independent work by other authors (Bergeron-Garsia-Haiman-Tesler and Knop-Lassalle-Okounkov-Sahi). The positivity conjectures are explicitly presented as open problems extending known conjectures, not as derived results. The manuscript is self-contained against external benchmarks with no evidence of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Based solely on the abstract, the paper introduces a new operator and new characters on top of standard symmetric function theory without providing independent evidence for the new objects.

axioms (1)
  • standard math Standard algebraic properties of symmetric functions and Macdonald polynomials hold as previously established.
    The paper builds directly on the theories initiated by Bergeron et al. and Knop et al.
invented entities (2)
  • Operator Γ no independent evidence
    purpose: To obtain a creation formula for Macdonald polynomials and connect two theories.
    Newly introduced in the paper; no independent evidence outside the derivation is mentioned.
  • Macdonald characters no independent evidence
    purpose: Two-parameter generalization of Jack characters.
    Defined via the new formula; no independent evidence outside the paper is mentioned.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A formula for the Jack super nabla operator

    math.CO 2025-09 unverdicted novelty 6.0

    A differential expression is established for the Jack analog of the super nabla operator via Chapuy-Dołęga and dehomogenized Nazarov-Sklyanin operators, derived from a general structure-coefficient operator G.

Reference graph

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34 extracted references · 34 canonical work pages · cited by 1 Pith paper

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