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arxiv: 2607.00940 · v1 · pith:JOLJOAY4new · submitted 2026-07-01 · 🧮 math.CO

The Schur positivity of nabla m_μ

Dun Qiu , Minhao Zhang This is my paper

Pith reviewed 2026-07-02 10:27 UTC · model grok-4.3

classification 🧮 math.CO
keywords nabla operatormonomial symmetric functionsSchur positivityLLT polynomialscompositional shuffle theoremssymmetric function positivityMacdonald polynomials
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The pith

The signed inner product of nabla applied to monomial symmetric functions with Schur functions has nonnegative coefficients in q and t.

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 partitions μ and λ of the same n, the polynomial (-1) to the power |μ| minus length of μ, times the inner product of the nabla operator on the monomial symmetric function m_μ with the Schur function s_λ, lies in the nonnegative integers adjoined with q and t. The result confirms a conjecture stated in 1999. The authors first establish a recursion that expands the signed monomial symmetric function as a linear combination with coefficients in nonnegative rationals in q of the symmetric functions C_α evaluated at 1. They then invoke the compositional shuffle theorems and the Schur positivity of LLT polynomials to finish the argument. The same recursion plus known positivity facts also yields the result for every positive integer power of nabla and an e-positive version after the change of variable q to q plus one.

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.

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).

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.

Where Pith is reading between the lines

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

  • 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.

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.

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.

read the original 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.

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 / 3 minor

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)
  1. 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.
  2. 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.
  3. 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

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

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.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The proof relies on standard background facts about the nabla operator and on two major external theorems; no new free parameters or invented entities are introduced.

axioms (3)
  • domain assumption Nabla acts diagonally on the modified Macdonald basis with eigenvalues involving q and t
    Invoked in the definition of ∇ and the statement of the conjecture.
  • domain assumption Compositional shuffle theorems of Carlsson-Mellit and Mellit hold
    Used to complete the proof after the recursion step.
  • domain assumption LLT polynomials are Schur positive
    Cited as a known result that supplies the final positivity.

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Reference graph

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