{"id":"94fb2c81-23dc-416b-90ff-3a17259d77c9","arxiv_id":"2607.06351","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The Schur positivity pattern of nabla^r G(k,n) depends exclusively on whether k divides n, resolving Bergeron's open problem.","lead":"The paper proves that the Schur positivity of the nabla operator applied to Petrie symmetric functions depends only on whether k divides n. This resolves an open problem in algebraic combinatorics about when certain symmetric function expansions have nonnegative coefficients.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified. The nonnegativity argument in Proposition 3.2 is condensed but complete upon careful verification.","rationale":"The reader correctly identified the most condensed part of the argument (Proposition 3.2's treatment of remaining terms), but characterized it as a potential gap when it is in fact a complete argument upon careful reading. The remaining terms are explicitly identifiable as the j-sum and the α_{T,b} terms with b < k, both of which manifestly have the form c(q) C_d(F̃_B) with c(q) ∈ N[q], and Lemma 2.6 handles the rest. No structural flaw exists. The paper's two main proofs (divisible and non-divisible cases) are verified: the algebra in Corollary 3.3 and Theorem 3.6 is correct, the nonnegativity of all coefficients is established, and the integrality + nonnegativity → N[q,t] argument is standard. The main residual risk is the dependency on very recent preprints [QZ26] and [BHM+25], but this is an external dependency, not an internal inconsistency. The reader's ACCEPT verdict with MODERATE confidence is appropriate; I would keep confidence at MODERATE given the dependency chain on recent results.","tokens_in":11091,"tokens_out":10339,"duration_ms":928401,"concrete_test":"Implement the Qiu-Zhang recursion (Lemma 2.5) in SageMath for small cases (e.g., k=3, a=2,3,4) and verify that the full expansion of (-1)^{ak} G(k,ak) in the C_α basis has all coefficients in Q_{≥0}[q], cross-checking against the explicit formula in Corollary 3.3. This would independently confirm that no remaining term in Proposition 3.2 produces a negative coefficient.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's concern targets Proposition 3.2, where the Qiu-Zhang recursion (Lemma 2.5) is applied to A = {k_1,...,k_{a-1}} with c = k, and the 'remaining terms' (after extracting the β_T and α_{T,k} contributions) are claimed to have nonnegative coefficients. Upon careful checking, the remaining terms fall into exactly two categories: (1) the j-sum Σ_{j=1}^{k-1} q^{j-1} C_j(F̃_{A∪{k-j}}), where q^{j-1} ∈ N[q] and Lemma 2.6 gives F̃_{A∪{k-j}} ∈ Σ N[q] C_α, so C_j of that is in N[q] C_{(j,α)}; (2) the α_{T,b} terms with 1 ≤ b ≤ k-1, where α_{T,b} ∈ N[q] by definition (2.12) and again F̃_{(A∖T)∪{b}} ∈ Σ N[q] C_α by Lemma 2.6. After division by a!, both land in Q_{≥0}[q] C_α. The β_T computation (β_T = (s+1)![k]_q since the inner sum q^k+...+q^{k-1} is empty when x=k) and the α_{T,k} computation (α_{T,k} = s! q^{k-1} since (x-1) mod x = x-1 = k-1) are both explicitly verified and correct. The subsequent Corollary 3.3 algebra ([k]_q = [k-1]_q + q^{k-1} causing exact cancellation of the q^{k-1} terms from Lemma 3.1) also checks out, yielding coefficients i/a · [k-1]_q ∈ Q_{≥0}[q]. For Theorem 1.2, Theorem 3.6's telescoping sum is verified step-by-step, and the non-divisibility condition ensures all C-indices are positive and the prefactor q^{n-k⌊n/k⌋-1} has nonnegative exponent. The dependency on recent preprints [QZ26, BHM+25] is a legitimate external risk, but the internal logic of the paper is sound.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","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.","tokens_in":11554,"tokens_out":966,"duration_ms":404444,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid and correct contribution. The main risk flagged by the reader—the condensed nonnegativity verification in Proposition 3.2—is, upon careful inspection, sound. The remaining terms do fall into the two categories the reader identified, and the application of Lemma 2.6 is valid. The recommendation of minor revision is appropriate to address the presentation issues, particularly the readability of the Proposition 3.2 verification. No concerns about novelty or scope."},"author_rebuttal":{"model":"glm-5.2","summary":"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.","responses":[],"tokens_in":10517,"tokens_out":106,"duration_ms":6049,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"This paper resolves Bergeron's open problem (recorded in Grinberg [Gri22]) on the Schur positivity pattern of nabla^r G(k,n). The main result: (-1)^{ak} nabla^r G(k,ak) is Schur positive when k divides n, and (-1)^{n-1} nabla^r G(k,n) is Schur positive when k does not divide n. Together these give a complete classification. This is a clean, correct result that closes a stated open problem, and the paper is short and to the point — 8 pages, no padding. The nondivisible case (Theorem 1.2) is the cleaner half. The key identity in Theorem 3.6 expresses (-1)^{n-1} G(k,n) as a sum of C_{n-ki} applied to signed monomial symmetric functions, with a nonnegative q-power prefactor. The telescoping argument is verified step by step, and non-divisibility ensures all C-indices are positive, so Proposition 2.4 applies directly. This part is solid. The divisible case (Theorem 1.1) is where the argument is most condensed. Proposition 3.2 applies the Qiu-Zhang recursion to a labeled multiset of a-1 copies of k, then collects terms. The beta_T and alpha_{T,k} computations are explicitly carried out and correct. But the claim that all remaining terms have nonnegative coefficients after division by a! is asserted in two sentences by noting they are of the form c(q) C_d(F_B) with c(q) in N[q] and invoking Lemma 2.6. I checked this: the remaining terms fall into two categories — the j-sum where q^{j-1} is in N[q] and Lemma 2.6 applies, and the alpha_{T,b} terms with b < k where alpha_{T,b} is in N[q] by definition. So the claim holds, but the paper would benefit from spelling this out rather than leaving it to the reader. This is a presentation gap, not a structural flaw. The dependency on [QZ26] (itself relying on [BHM+25]) is worth noting — if that chain has issues, this paper inherits them. But the internal logic is sound. The Corollary 3.3 step using [k]_q = [k-1]_q + q^{k-1} to get exact cancellation and land coefficients in Q_{>=0}[q] is correct. This paper is for algebraic combinatorists working on the nabla operator program. It deserves a serious referee who can verify the condensed nonnegativity argument in Proposition 3.2 line by line. I recommend accepting for peer review.","headline":"Resolves Bergeron's open problem on Schur positivity of nabla on Petrie symmetric functions; proofs check out with one condensed step.","tokens_in":12237,"tokens_out":650,"would_cite":true,"duration_ms":225873,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Sign pattern for nabla on Petrie functions pinned to divisibility test","keywords":[],"falsifier":"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.","tokens_in":11386,"feed_emoji":"🔧","tokens_out":1107,"duration_ms":124708,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sign pattern for nabla on Petrie functions pinned to divisibility test","feed_subtitle":"Whether k divides n decides if nabla^r G(k,n) is Schur positive, resolving Bergeron's open conjecture","key_machinery":"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.","core_discovery":"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.","pith_inferences":[],"forward_implications":["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."],"fun_headline_variants":["Divisibility by k decides Schur positivity of nabla on Petrie functions","Bergeron conjecture resolved: k|n governs nabla sign on Petrie G(k,n)","Nabla on Petrie G(k,n) is Schur positive iff k divides n","Full Schur positivity pattern for nabla on Petrie functions found"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Divisibility by k decides Schur positivity of nabla on Petrie functions","Bergeron conjecture resolved: k|n governs nabla sign on Petrie G(k,n)","Nabla on Petrie G(k,n) is Schur positive iff k divides n","Full Schur positivity pattern for nabla on Petrie functions found"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":561,"prompt_tokens":485,"completion_tokens":76,"prompt_tokens_details":null},"tokens_in":485,"tokens_out":76,"duration_ms":18164,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T08:49:02.310868+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"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.","supporting_citations":[],"review_version":1}