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Schur positivity of nabla on Petrie symmetric functions

T0 review · 0 major / 4 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Sign pattern for nabla on Petrie functions pinned to divisibility test

desk verdict Resolves Bergeron's open problem on Schur positivity of nabla on Petrie symmetric functions; proofs check out with one condensed step. read the letter →

arxiv 2607.06351 v1 pith:A474QERU submitted 2026-07-07 math.CO

Menghao Qu This is my paper · ORCID
classification math.CO
keywords lambdasymmetricfunctionsgrinbergnablapetriepositivityschur
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

The Petrie symmetric function G(k,n) is the sum of all monomial symmetric functions indexed by partitions of n whose largest part is strictly less than k. This paper determines exactly when applying the nabla operator (any number of times) to G(k,n) yields a Schur-positive symmetric function—meaning one whose Schur basis expansion has nonnegative polynomial coefficients. The answer depends entirely on whether k divides n. When k divides n (so n = ak), the signed function (-1)^{ak} nabla^r G(k,ak) is Schur positive. When k does not divide n, (-1)^{n-1} nabla^r G(k,n) is Schur positive. Together these two cases give a complete classification, resolving a conjecture of Bergeron recorded in Grinberg's work. The proof splits along the same divisibility line. In the divisible case, the author combines a prior identity (joint with Xin) expressing (-1)^{ak} G(k,ak) in terms of creation operators C_a acting on signed monomial symmetric functions with a recent recursive framework of Qiu and Zhang, showing that the resulting C_alpha-expansion has nonnegative rational coefficients. In the nondivisible case, the author derives a new identity expressing (-1)^{n-1} G(k,n) as a sum of creation operators C_{n-ki} applied to signed monomial symmetric functions indexed by rectangular partitions; since nabla^r preserves Schur positivity on each C_alpha, the result follows directly.

What carries the argument

The Haglund-Morse-Zabrocki creation operators C_a, which act on symmetric functions and satisfy nabla^r C_alpha being Schur positive for any composition alpha. The Qiu-Zhang recursion (Lemma 2.5), which expands signed monomial symmetric functions F_A associated to labeled multisets into C_alpha-components with controlled coefficients. Grinberg's identity (Lemma 3.5) expressing G(k,n) as an alternating sum of products m_{k^i} h_{n-ki}. A new identity (Theorem 3.6) converting this alternating sum into a clean C-operator expansion.

What would settle it

A single counterexample: any k, n, r, and Schur index lambda for which the predicted sign fails to yield a polynomial with nonnegative coefficients. Concretely, if k divides n=ak but (-1)^{ak} nabla^r G(k,ak) has a negative Schur coefficient for some r and lambda, or if k does not divide n but (-1)^{n-1} nabla^r G(k,n) does, the classification is wrong.

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Extended reading notes

Core claim

The complete sign pattern for Schur positivity of nabla^r G(k,n) is governed solely by whether k divides n: if k|n then (-1)^{ak} nabla^r G(k,ak) is Schur positive, and if k does not divide n then (-1)^{n-1} nabla^r G(k,n) is Schur positive. The nondivisible case rests on a new identity (Theorem 3.6) expressing (-1)^{n-1} G(k,n) as a positively-weighted sum of C_{n-ki} applied to signed monomial functions m_{k^i}, which reduces Schur positivity to the known positivity of nabla on each C_alpha. The divisible case uses the Qiu-Zhang recursion on a labeled multiset of copies of k to extract a nonnegative C_alpha-expansion.

Load-bearing premise

In the divisible case, the proof applies the Qiu-Zhang recursion to a specific labeled multiset and asserts that all terms beyond those explicitly collected have nonnegative rational coefficients after division by a factorial, based on their structural form. The detailed verification that every term from the recursion satisfies this nonnegativity is condensed rather than spelled out term by term.

Editorial extensions

If this is right

  • The sign (-1)^{ak} vs (-1)^{n-1} provides a testable divisibility criterion: for any specific k and n, one can predict the sign of every Schur coefficient of nabla^r G(k,n) without computing the expansion.
  • The new identity in Theorem 3.6 gives an explicit C_alpha-expansion of G(k,n) for k not dividing n, which could serve as a starting point for combinatorial interpretations (e.g., parking function models) of these symmetric functions.
  • The breakdown at k|n (where the C_0 operator appears and positivity fails) identifies a structural boundary that may guide the search for refined or compositional versions of these positivity results.
  • The reliance on the Qiu-Zhang recursion suggests the same recursive framework could be applied to other families of symmetric functions built from monomial sums with part-size restrictions.
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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 / 4 minor

Summary. The paper resolves an open problem of Bergeron (recorded in [Gri22, Conjecture 5.2]) concerning the Schur positivity of $abla^r G(k,n)$, where $G(k,n)$ is the Petrie symmetric function and $abla$ is the Bergeron-Garsia nabla operator. The main results, Theorem 1.1 and Theorem 1.2, completely determine the sign pattern: $(-1)^{ak} abla^r G(k,ak)$ is Schur positive when $k mid n$, and $(-1)^{n-1} abla^r G(k,n)$ is Schur positive when $k mid n$, for all $r geq 1$. The proof splits into two cases. The divisible case (Section 3.1) combines an identity from prior joint work with Xin [QX25] with the Qiu-Zhang recursion [QZ26] to extract a nonnegative $C_alpha$-expansion. The nondivisible case (Section 3.2) uses Grinberg's identity for $G(k,n)$ and a computation with creation operators to express $(-1)^{n-1}G(k,n)$ directly as a positive sum of $C_alpha$ applied to signed monomial symmetric functions.

Significance. The resolution of [Gri22, Conjecture 5.2] is a notable contribution to algebraic combinatorics. The paper builds on recent and active developments, including the Qiu-Zhang recursion and the proof of the Loehr-Warrington conjecture by Blasiak et al. [BHM+25]. The derivation in the nondivisible case (Theorem 3.6) is notably clean and parameter-free, yielding a direct telescoping identity. The divisible case successfully isolates nonnegative coefficients from a complex recursion. The results provide a complete and falsifiable classification of the sign pattern for the nabla operator on the Petrie basis.

minor comments (4)
  1. In the proof of Proposition 3.2, the verification that the 'remaining terms' (the $j$-sum and the $alpha_{T,b}$ terms with $b < k$) have nonnegative coefficients in $mathbb{Q}_{geq 0}[q]$ after division by $a!$ is condensed into a single sentence. While the argument invoking Lemma 2.6 is correct, expanding this verification by one or two sentences to explicitly note that the $j$-sum terms are of the form $q^{j-1} C_j( ilde{F}_{A cup {k-j}})$ with $q^{j-1} in mathbb{N}[q]$, and that the $alpha_{T,b}$ terms have $alpha_{T,b} in mathbb{N}[q]$ by definition (2.12), would improve readability and allow the reader to verify the nonnegativity without reconstructing the argument.
  2. In Theorem 3.6, the telescoping computation is presented in full detail, which is helpful. However, the exchange of summation order in the second term could benefit from a brief justification that the sum is finite, ensuring no convergence issues arise in the formal power series context.
  3. The paper cites several recent preprints ([QZ26], [BHM+25], [Qu26], [KO24]). The reference for [BHM+25] indicates publication in J. Amer. Math. Soc. (2025), but others are listed as preprints. The author should ensure that all references are updated to their most current published versions at the time of final submission.
  4. In the introduction, the phrase 'To the best of our knowledge, a proof of the Schur positivity of $abla s_lambda$, up to a sign, is still lacking' could be clarified. Given that the paper cites [BHM+25] and [KO24] for the monomial expansion, a brief indication of what is known versus what is open would help the reader contextualize the state of the art.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and the positive assessment. The referee's summary accurately describes the structure and content of the paper, and we appreciate the recognition of the clean telescoping argument in the nondivisible case and the extraction of nonnegative coefficients in the divisible case.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation chain is self-contained against external benchmarks

full rationale

The paper proves Schur positivity of signed nabla^r G(k,n) by combining (1) a symmetric function identity from [QX25] (Lemma 3.1), (2) the Qiu-Zhang recursion [QZ26] (Lemma 2.5), and (3) Grinberg's identity [Gri22] (Lemma 3.5). The self-citation to [QX25] (joint with Xin) provides Lemma 3.1, which is a parameter-free algebraic identity — not a fitted input renamed as a prediction. The Qiu-Zhang recursion [QZ26] is independently derived from the Loehr-Warrington conjecture proved by Blasiak et al. [BHM+25], which is external to the author. The main derivation in Proposition 3.2 applies this recursion to a specific labeled multiset and verifies, through explicit computation of beta_T and alpha_{T,k} terms, that all coefficients land in Q_{>=0}[q]. Theorem 3.6's telescoping sum is verified step-by-step from Lemma 3.4 (proved in-house via plethystic calculus) and Lemma 3.5 (Grinberg's identity). No step reduces to its inputs by construction; no fitted parameters are renamed as predictions; no ansatz is smuggled via self-citation. The self-citation to [QX25] is a load-bearing identity, but it is a parameter-free symmetric function identity that is independently verifiable and does not include the target result (Schur positivity) among its assumptions. The derivation is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are introduced; all results are parameter-free algebraic identities. No invented entities. The axioms are established results from the prior literature, each with independent verification (published or preprint with traceable proofs). The paper builds on the Qiu-Zhang framework and Grinberg's identity without postulating new structures.

assumptions (4)
  • domain assumption Schur positivity of nabla^r C_alpha for r>=1 (Proposition 2.4), derived from the compositional shuffle theorem and Schur positivity of LLT polynomials
    Section 2.2, Proposition 2.4. This is the foundational result ensuring nabla^r preserves Schur positivity on C_alpha. Established by Carlsson-Mellit [CM18] and Grojnowski-Haiman [GH07].
  • domain assumption Qiu-Zhang recursion (Lemma 2.5), providing the recursive expansion of signed monomial symmetric functions in terms of creation operators
    Section 2.3, Lemma 2.5. The central tool for the divisible case. Proved in [QZ26] using the Loehr-Warrington conjecture.
  • standard math Grinberg's identity G(k,n) = sum (-1)^i m_{k^i} h_{n-ki} (Lemma 3.5)
    Section 3.2, Lemma 3.5, from [Gri22, Theorem 2.19]. Used to express G(k,n) in terms of rectangular monomials and homogeneous symmetric functions.
  • domain assumption Integrality of nabla^r G(k,n) Schur coefficients in Z[q,t] (from Lemma 2.8)
    Section 3, opening paragraph. Reduces the positivity problem to showing nonnegativity of C_alpha-expansion coefficients. Derived from [QZ26].

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Pith. "Pith review of Schur positivity of nabla on Petrie symmetric functions." pith.science (2026). https://pith.science/paper/A474QERU

@misc{pith2026260706351,
  author       = {Pith},
  title        = {Pith review of: Schur positivity of nabla on Petrie symmetric functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A474QERU}},
  note         = {Machine review of arXiv:2607.06351}
}
abstract

The Petrie symmetric function $G(k,n)$, introduced by Grinberg, is defined as the sum of monomial symmetric functions $m_\lambda$ indexed by partitions $\lambda\vdash n$ satisfying $\lambda_1<k$. This article demonstrates that the Schur positivity pattern of $\nabla^r G(k,n)$ for all $r\geq 1$ depends exclusively on whether $k$ divides $n$, thus answering an open problem of Bergeron noted in Grinberg's work.

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Works this paper leans on

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