REVIEW 3 minor 1 cited by
The Schur positivity of $\nabla m_\mu$
T0 review · 0 major / 3 minor · reviewed 2026-07-02 · grok-4.3
Pith's one-line read The signed inner product of nabla applied to monomial symmetric functions with Schur functions has nonnegative coefficients in q and t.
desk verdict Qiu and Zhang give a recursion proof of the 1999 conjecture that signed nabla on monomials is Schur positive. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The recursion that expresses (-1)^{|μ|-ℓ(μ)} m_μ as a nonnegative linear combination in ℚ≥0[q] of the symmetric functions C_α(1).
What would settle it
An explicit computation for any small n and partitions μ,λ of n that produces a negative coefficient in (-1)^{|μ|-ℓ(μ)} ⟨∇ m_μ, s_λ⟩ would disprove the claim.
Extended reading notes
Core claim
The central claim is that (-1)^{|μ|-ℓ(μ)} ⟨∇ m_μ, s_λ⟩ belongs to ℕ[q,t] for every pair of partitions μ,λ of n. The proof rests on a recursion that writes (-1)^{|μ|-ℓ(μ)} m_μ as a sum with coefficients in ℚ≥0[q] of the symmetric functions C_α(1), followed by application of the compositional shuffle theorems of Carlsson-Mellit and Mellit together with Schur positivity of LLT polynomials; the identical method gives the same positivity for ∇^r m_μ when r ≥ 1 and an e-positive analogue after q ↦ q+1.
Load-bearing premise
The recursion that writes the signed monomial symmetric function as a nonnegative combination of the C_α(1) functions must hold and remain compatible with the compositional shuffle theorems and the Schur positivity of LLT polynomials.
Editorial extensions
If this is right
- The same positivity holds for every positive integer power r of the nabla operator applied to m_μ.
- An e-positive analogue holds after the substitution q maps to q plus one, using the e-positivity of column LLT polynomials.
- The recursion provides a uniform way to obtain positivity statements for other diagonal operators once compatible positivity theorems for LLT polynomials are available.
Reading between the lines
- The recursion may be iterable to produce explicit combinatorial formulas or generating functions for the coefficients.
- Similar expansions could be tested on other families of symmetric functions to obtain new positivity results for nabla or related operators.
- The method suggests that proving recursions into C_α(1) with controlled coefficients is a viable route for attacking other open positivity conjectures in the theory of Macdonald polynomials.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves the 1999 Bergeron-Garsia-Haiman-Tesler conjecture that (-1)^{|μ|-ℓ(μ)} ⟨∇ m_μ, s_λ⟩ lies in ℕ[q,t] for all partitions μ,λ ⊢ n. It establishes a recursion expressing (-1)^{|μ|-ℓ(μ)} m_μ as a ℚ≥0[q]-linear combination of the C_α(1) basis elements, then invokes the compositional shuffle theorems of Carlsson-Mellit and Mellit together with Schur positivity of LLT polynomials to obtain the result. The argument extends verbatim to ∇^r for r ≥ 1 and yields an e-positive analogue after the substitution q ↦ q+1.
Significance. If the recursion and its compatibility with the cited shuffle and LLT results hold, the paper resolves a long-standing open problem in Macdonald theory and diagonal harmonics. The method supplies an explicit recursive reduction to known positivity statements rather than a direct combinatorial interpretation, and the same framework produces the e-positive variant without additional machinery.
minor comments (3)
- The recursion is introduced in the abstract but its precise statement (including the range of α and the base cases) should be highlighted with an equation number in §2 or §3 for quick reference.
- In the paragraph combining the recursion with the shuffle theorems, add a one-sentence reminder of the exact form of the C_α(1) expansion that is fed into the shuffle operator.
- The e-positive analogue is stated only in the abstract; a short dedicated paragraph or corollary after the main proof would improve readability.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript and for recommending acceptance.
Circularity Check
No significant circularity; derivation relies on independent external theorems
full rationale
The paper establishes its own recursion expressing (-1)^{|μ|-ℓ(μ)} m_μ as a nonnegative combination in the C_α(1) basis over Q≥0[q], then combines this with the compositional shuffle theorems of Carlsson-Mellit and Mellit plus Schur positivity of LLT polynomials. These supporting results are from independent prior work by other authors, are not derived from the target positivity statement, and are externally verifiable. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain.
Assumptions & free parameters
assumptions (3)
- domain assumption Nabla acts diagonally on the modified Macdonald basis with eigenvalues involving q and t
- domain assumption Compositional shuffle theorems of Carlsson-Mellit and Mellit hold
- domain assumption LLT polynomials are Schur positive
Cite this review
Pith. "Pith review of The Schur positivity of $\nabla m_\mu$." pith.science (2026). https://pith.science/paper/JOLJOAY4
@misc{pith2026260700940,
author = {Pith},
title = {Pith review of: The Schur positivity of $\nabla m_\mu$},
year = {2026},
howpublished = {\url{https://pith.science/paper/JOLJOAY4}},
note = {Machine review of arXiv:2607.00940}
}
abstract
Bergeron, Garsia, Haiman and Tesler conjectured in 1999 that, for all partitions $\mu,\lambda\vdash n$, the polynomial $(-1)^{|\mu|-\ell(\mu)}\langle \nabla m_\mu, s_\lambda\rangle$ has nonnegative integer coefficients, where $\nabla$ is the Bergeron--Garsia nabla operator, which acts diagonally on the modified Macdonald basis, and $m_\mu$ is the monomial symmetric function. In this article, we prove this conjecture, and more generally that $(-1)^{|\mu|-\ell(\mu)}\langle\nabla^r m_\mu,s_\lambda\rangle\in\mathbb{N}[q,t]$ for all $r\geq 1$. We establish a recursion showing that $(-1)^{|\mu|-\ell(\mu)}m_\mu$ has an expansion with coefficients in $\mathbb{Q}_{\geq 0}[q]$ in the symmetric functions $C_\alpha(1)$, where $C_a$ denotes the operator introduced by Haglund, Morse and Zabrocki. Combining this expansion with the compositional shuffle theorems of Carlsson--Mellit and Mellit, and with the Schur positivity of LLT polynomials, completes the proof. The same method, using the $e$-positivity of column LLT polynomials after the substitution $q\mapsto q+1$, also gives an $e$-positive analogue.
Forward citations
Cited by 1 Pith paper
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Schur positivity of nabla on Petrie symmetric functions
The Schur positivity pattern of nabla^r G(k,n) depends exclusively on whether k divides n, resolving Bergeron's open problem.
Reviewed July 2, 2026 · model on record in the stance chip above.
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