{"id":"142a2a27-3769-4bf3-bf6c-71deaeff2afb","arxiv_id":"2607.00940","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For all partitions mu, lambda and all r >= 1, (-1)^(|mu|-ell(mu)) <nabla^r m_mu, s_lambda> is in N[q,t], resolving the 1999 Bergeron-Garsia-Haiman-Tesler conjecture.","lead":"This paper proves a 1999 conjecture in algebraic combinatorics: a signed, two-variable version of an operator applied to monomial symmetric functions always expands into Schur functions with nonnegative integer coefficients. The proof builds a new recursion for eF_mu in the C basis and combines it with shuffle theorems and LLT positivity, also yielding an e-positive analogue.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.2 for arbitrary r depends on the imported integrality statement (1), which is not proved here; if [8] only covers r=1, the general-r claim is unsupported.","rationale":"The central proof is a long but apparently correct induction. The C-expansion Theorem 4.2 is internally consistent and independently supported by SageMath for all partitions of size at most 12. I checked the plethystic signs in Lemmas 2.1 and 2.2 under the standard alphabet-difference convention, and the recursion in Lemmas 4.5--4.8 appears sound. The only genuinely load-bearing assumption I find is the imported integrality statement (1): without it, the conclusion of Theorem 1.2 is only rational Schur positivity, not integer Schur positivity. Since Theorem 1.2 is stated for all r>=1, and the scope of the cited result in [8] is not reproduced in the paper, acceptance should be conditional on confirming that [8] indeed proves the needed LLT expansion for all r. For r=1, the integrality is already in [4], so the original Bergeron--Garsia--Haiman--Tesler conjecture is not endangered even if the general-r integrality check fails.","tokens_in":20010,"tokens_out":38378,"duration_ms":292049,"concrete_test":"Verify the exact statement of the main theorem of [8]: does it provide an LLT expansion of nabla^r s_lambda with coefficients in Z[q,t] for every integer r>=1, or only for r=1? As an independent numerical check, compute (1) for r=2,3 and all partitions mu,lambda of n with n<=6 by expanding nabla^r m_mu in the modified Macdonald eigenbasis and using the integer inverse Kostka matrix; if any coefficient is non-integral, the premise (1) is false.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.2 is obtained by combining two facts: Theorem 4.2 plus Proposition 2.3 give (-1)^{|mu|-ell(mu)} <nabla^r m_mu, s_lambda> in Q_{\\ge 0}[q,t], while statement (1) in the final paragraph of Section 1 asserts the same polynomial lies in Z[q,t]. The implication to N[q,t] is valid, so (1) is the only step that upgrades rational nonnegativity to integrality. Statement (1) is derived from [8] (an LLT expansion of nabla^r s_lambda with integral coefficients) and [13] (integral inverse Kostka matrix). If [8]'s Loehr--Warrington theorem is only for r=1, then (1) is not established for r>=2 and the proof of the full Theorem 1.2 collapses to the r=1 case, which still resolves Conjecture 1.1 since [4] contains integrality for r=1. No internal algebraic error was found in the C-operator recursion; the SageMath check in Remark 4.4 supports Theorem 4.2, but it does not test (1).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves the 1999 Bergeron–Garsia–Haiman–Tesler conjecture that (-1)^{|mu|-ell(mu)} <nabla m_mu, s_lambda> has nonnegative integer coefficients, and extends it to nabla^r for all r>=1. The proof introduces labelled multisets and the symmetric functions eF_A, proves a recursion for the quantity Theta_{A,c} (Theorem 4.1), derives a C-expansion of eF_A with coefficients in N[q] (Theorem 4.2), and then combines this expansion with compositional (km,kn)-shuffle theorems and LLT positivity to obtain Schur positivity. The same strategy yields an e-positive analogue (Theorem 1.3) and an extension to the Schiffmann algebra (Theorem 5.1).","tokens_in":20226,"tokens_out":21074,"duration_ms":168924,"significance":"If correct, the paper resolves a long-standing conjecture in a strong form and supplies a uniform recursive method for signed monomial symmetric functions under nabla. The combinatorial recursion is worked out in detail, the proof is self-contained apart from clearly identified external inputs, and the authors report symbolic SageMath verification for all partitions of size at most 12 in Remark 4.4, which is a meaningful check. The main novelty is the new recursion and the resulting positive C-expansion, which is the kind of structural input that should be reusable beyond the specific theorem.","major_comments":[{"comment":"The proof of Theorem 1.2 for r>=2 rests entirely on the integrality assertion (1). The manuscript says that (1) follows from [8] and [13], but it does not identify the theorem in [8] that provides an LLT expansion of nabla^r s_lambda for every r>=1. If [8] treats only r=1, then the general-r claim is unsupported and Theorem 1.2 would need to be restricted to r=1; the r=1 case is unaffected because [4] already contains the needed integrality. Please supply the precise statement and proof location in [8] (or prove (1) directly), or weaken the theorem and abstract accordingly. This is the only load-bearing gap I found; the recursive C-expansion argument itself appears sound.","section":"Section 1, Eq. (1); proof of Theorem 1.2, Section 4.1"}],"minor_comments":[{"comment":"The phrase 'and then!-theorem' appears to be a typo for 'and n!-theorem'.","section":"Section 1, paragraph 1"},{"comment":"The inner summation bound 'b=S_S+1' is rendered with the subscript omitted in places; please ensure the notation S_S is typeset uniformly.","section":"Section 4.2, around Eq. (23)"},{"comment":"The sentence 'ourproofexpresses' in the first paragraph of Section 5.2 is missing a space and should read 'our proof expresses'.","section":"Section 5.2, Conjectures"}],"recommendation":"major_revision","confidential_remarks":"The only substantive concern is the provenance of (1) for r>=2. If the authors can supply a precise reference or a short proof of that integrality statement, the paper would be suitable for acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper resolves the 1999 Bergeron–Garsia–Haiman–Tesler conjecture and, more generally, proves that (-1)^{|mu|-ell(mu)} <nabla^r m_mu, s_lambda> lies in N[q,t] for every r >= 1. That is a real breakthrough: the best prior results covered only special partition shapes (e_n, p_n, hooks, (2^k,1^ell)), and the general case had been open for over two decades. The main new engine is a labelled-multiset recursion, Theorem 4.1, which yields a C_alpha(1)-expansion with coefficients in N[q] for the augmented monomial functions eF_A (Theorem 4.2). The recursion appears to be genuinely new, not a routine adaptation, because it has to handle arbitrary multiplicities. I read the operator identities and the induction in Section 4 closely; they check out. The SageMath verification for |mu| <= 12 is a useful sanity check, though not a proof.\n\nThe main soft spot is the integrality input. To go from nonnegative rational coefficients to N[q,t] in Theorem 1.2, the paper invokes statement (1): that (-1)^{|mu|-ell(mu)} <nabla^r m_mu, s_lambda> in Z[q,t] for all r. For r=1 this is a result of Bergeron–Garsia–Haiman–Tesler themselves. For r>=2 the paper attributes it to Blasiak–Haiman–Morse–Pun–Seelinger [8] plus Eğecioğlu–Remmel [13]. I cannot inspect [8] directly, so I cannot independently confirm that the Loehr–Warrington theorem proved there covers every r. The stress-test note entertains the possibility that it only covers r=1. If so, Theorem 1.2 for r>=2 would collapse to the r=1 case (which still settles the 1999 conjecture), and the Schiffmann-algebra statements in Section 5 would also need adjustment. But the paper states the attribution as a fact, and nothing internal suggests an error. This is the one thing I would want a referee to verify before publication: ask for the precise theorem number in [8] that implies integrality for all r.\n\nThe other caveat is minor: the proved C-expansion of eF_A has N[q] coefficients, but after dividing by prod m_i(mu)! the expansion of F_mu itself has rational coefficients. The paper flags this and makes Conjecture 5.2 about integrality. That is honest and clearly stated.\n\nBottom line: the central argument holds up, the paper is carefully written, and the result is important. It deserves a serious referee. I would send it out with instructions to check the exact scope of [8]'s Loehr–Warrington theorem; if that citation covers all r, this is an accept with no substantive revision needed. For researchers in algebraic combinatorics, especially those working on nabla, LLT polynomials, and diagonal harmonics, this will be a standard reference.","headline":"A strong paper that resolves the 1999 BGHT conjecture and extends it to all powers of nabla; the proof is sound, but the r>=2 integrality step rests on a citation whose scope should be verified.","tokens_in":854,"tokens_out":1475,"would_cite":true,"duration_ms":50247,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10","20C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for every pair of partitions μ, λ of n and every integer r ≥ 1, the signed inner product (-1)^{|μ|-ℓ(μ)}⟨∇^r m_μ, s_λ⟩ is a polynomial with nonnegative integer coefficients, settling a 1999 conjecture of Bergeron…","keywords":["symmetric functions","Schur positivity","nabla operator","monomial symmetric functions","C operators","LLT polynomials","Macdonald polynomials","shuffle theorem"],"falsifier":"Computing $(-1)^{|\\mu|-\\ell(\\mu)}\\langle\\nabla^r m_\\mu,s_\\lambda\\rangle$ for a concrete case such as $\\mu=\\lambda=(3,3), r=1$ via the recursion of Theorem 4.2 and the shuffle theorem and checking that every monomial coefficient is a nonnegative integer would settle the claim, since any negative coefficient would refute it (and the paper's computational check up to $|\\mu|=12$ finds none).","tokens_in":19802,"feed_emoji":"➕","tokens_out":12762,"duration_ms":91381,"temperature":0.7,"pith_summary":"This paper proves a 1999 conjecture of Bergeron, Garsia, Haiman and Tesler: for every pair of partitions $\\mu,\\lambda$ of $n$, the polynomial $(-1)^{|\\mu|-\\ell(\\mu)}\\langle\\nabla m_\\mu,s_\\lambda\\rangle$ in the two variables $q,t$ has only nonnegative integer coefficients. The result is stronger, covering every power of the nabla operator: $(-1)^{|\\mu|-\\ell(\\mu)}\\langle\\nabla^r m_\\mu,s_\\lambda\\rangle\\in\\mathbb{N}[q,t]$ for all $r\\ge 1$. The proof gives a recursion that expands the signed monomial function in a positive rational way using the $C_\\alpha(1)$ symmetric functions, then upgrades the resulting nonnegative rational coefficients to nonnegative integers using a separately established integrality theorem. The same machinery produces an $e$-positive companion statement after the substitution $q\\mapsto q+1$ and extends to the Schiffmann algebra.","feed_headline":"Signed nabla monomials are Schur positive","feed_subtitle":"Settles the Bergeron–Garsia–Haiman–Tesler conjecture for all partitions and all powers r ≥ 1","key_machinery":"The engine of the proof is the recursion of Theorem 4.1 for $\\Theta_{A,c}=eF_{A\\cup\\{c\\}}-\\sum_{j=1}^{c-1}q^{j-1}C_j(eF_{A\\cup\\{c-j\\}})$, which expands $\\Theta_{A,c}$ in the $C_\\alpha(1)$-like functions $C_{c+S_T-b}(eF_{(A\\setminus T)\\cup\\{b\\}})$ and $C_{c+S_T}(eF_{A\\setminus T})$ with explicit coefficients $\\alpha_{T,b},\\beta_T\\in\\mathbb{N}[q]$ (equations (18)\\textendash(19)). The $C_a$ operators, introduced by Haglund, Morse and Zabrocki, send $C_\\alpha(1)$ to $C_{(a,\\alpha_1,\\dots)}(1)$, so the recursion iterates to write every $eF_A$ as an $\\mathbb{N}[q]$-combination of the $C_\\alpha(1)$. The proof rests on the commutation identity of Lemma 3.4 (moving $eF_B$ past $C_r$), the product identity of Lemma 3.5, and the coefficient evaluations of Lemmas 4.6\\textendash 4.8, together with the augmented brick-tabloid formula of Lemma 3.2.","core_discovery":"The paper's central discovery is a positivity machine for the monomial basis under the nabla operator. For a partition $\\mu$, write $F_\\mu=(-1)^{|\\mu|-\\ell(\\mu)}m_\\mu$ and $eF_\\mu=(\\prod_i m_i(\\mu)!)F_\\mu$; the augmented function $eF_\\mu$ is shown (Theorem 4.2) to have an expansion $\\sum_\\alpha b_{\\mu,\\alpha}C_\\alpha(1)$ with every $b_{\\mu,\\alpha}\\in\\mathbb{N}[q]$, produced by an explicit recursion (Theorem 4.1) on labelled multisets. After dividing by the multiplicity factor, $F_\\mu$ becomes a $\\mathbb{Q}_{\\ge0}[q]$-linear combination of the $C_\\alpha(1)$. Mellit's compositional $(km,kn)$-shuffle theorem writes $\\nabla^r C_\\alpha(1)$ as a nonnegative integer sum of column LLT polynomials, and LLT polynomials are Schur positive by Grojnowski\\textendash Haiman, so each $\\langle\\nabla^r C_\\alpha(1),s_\\lambda\\rangle$ lies in $\\mathbb{N}[q,t]$. Consequently $(-1)^{|\\mu|-\\ell(\\mu)}\\langle\\nabla^r m_\\mu,s_\\lambda\\rangle$ is a nonnegative rational polynomial; the imported integrality statement (1) then forces it into $\\mathbb{N}[q,t]$, proving Theorem 1.2 and resolving Conjecture 1.1.","pith_inferences":["Because $C$-expansions are not unique, the rational-coefficient expansion used in the proof is just one of many; a canonical positive $C$-expansion of $F_\\mu$ (the goal of Conjecture 5.2) could plausibly encode a parking-function statistic for $\\nabla^r m_\\mu$, extending the known hook and two-row cases.","The paper demonstrates a reusable template\\textemdash rational Schur positivity from shuffle-plus-LLT machinery, upgraded to integer positivity by an independent integrality theorem\\textemdash that could certify Schur positivity for other signed monomial-like families whenever a matching integrality result exists.","The $e$-positive companion after $q\\mapsto q+1$ suggests probing other plethystic substitutions in $q$: each substitution that preserves positivity of column LLT polynomials would yield a new family of $e$-positive images of $\\nabla^r m_\\mu$.","The Schiffmann-algebra extension indicates the positivity is not special to $\\nabla$ itself but is shared by the whole family of diagonal operators attached to coprime $(r,k)$; this hints the conjecture generalizes further in the direction of the Schiffmann algebra literature."],"forward_implications":["Conjecture IV of Bergeron\\textendash Garsia\\textendash Haiman\\textendash Tesler (1999) becomes a theorem: the signed nabla of each monomial symmetric function is Schur positive in $\\mathbb{N}[q,t]$.","The higher-power statement holds: $(-1)^{|\\mu|-\\ell(\\mu)}\\langle\\nabla^r m_\\mu,s_\\lambda\\rangle\\in\\mathbb{N}[q,t]$ for every $r\\ge 1$ and every pair of partitions $\\mu,\\lambda$.","An $e$-positive companion follows after the substitution $q\\mapsto q+1$: $(-1)^{|\\mu|-\\ell(\\mu)}\\langle(\\nabla^r m_\\mu)[X;q+1],\\omega(m_\\lambda)\\rangle\\in\\mathbb{N}[q,t]$ (Theorem 1.3).","The same $C$-expansion extends to the Schiffmann algebra: $(-1)^{|\\mu|-\\ell(\\mu)}\\langle m_\\mu[-MX_{r,k}]\\cdot 1,s_\\lambda\\rangle\\in\\mathbb{N}[q,t]$ for coprime positive integers $r,k$ (Theorem 5.1).","The recursion is effective and was checked computationally for all $|\\mu|\\le 12$; if Conjecture 5.2 holds, then $(-1)^{|\\mu|-\\ell(\\mu)}\\nabla^r m_\\mu$ would itself be a nonnegative integer combination of column LLT polynomials, refining Schur positivity to LLT positivity."],"supporting_citations":[{"why":"formulated the conjecture and proved the integrality of the signed inner product for $r=1$.","marker":"[4]"},{"why":"proved the compositional $(km,kn)$-shuffle theorem that expresses $\\nabla^r C_\\alpha(1)$ as a nonnegative sum of column LLT polynomials.","marker":"[33]"},{"why":"established Schur positivity of LLT polynomials, converting the LLT sum into Schur positivity.","marker":"[20]"},{"why":"gave an LLT expansion of $\\nabla^r s_\\lambda$ with integral coefficients, a key ingredient in the integrality statement (1) for all $r$.","marker":"[8]"},{"why":"proved integrality of the inverse Kostka matrix, used with [8] to obtain the integrality statement (1).","marker":"[13]"},{"why":"introduced the $C_a$ operators and the basic identities (5)–(7) on which the entire recursion rests.","marker":"[23]"},{"why":"proved $e$-positivity of column LLT polynomials after $q\\mapsto q+1$, yielding Theorem 1.3.","marker":"[12]"},{"why":"gave the brick-tabloid transition formula used in Lemma 3.2 for the augmented monomial functions.","marker":"[14]"},{"why":"proved the compositional shuffle theorem, the $r=1$ case that sets the pattern for the higher-power shuffle input.","marker":"[11]"}],"fun_headline_variants":["Bergeron-Garsia-Haiman-Tesler nabla conjecture proved","Signed nabla monomials are Schur positive, all r","Nabla on monomial basis: Schur positivity resolved","1999 conjecture on nabla monomials proven","Schur positivity for nabla monomials, all powers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on a previously established fact that the signed coefficient polynomial is an integer polynomial; without that fact, the method would certify only nonnegative rational coefficients.","fun_headline_variants_meta":{"raw":{"variants":["Bergeron-Garsia-Haiman-Tesler nabla conjecture proved","Signed nabla monomials are Schur positive, all r","Nabla on monomial basis: Schur positivity resolved","1999 conjecture on nabla monomials proven","Schur positivity for nabla monomials, all powers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000501,"raw_usage":{"total_tokens":2545,"prompt_tokens":1134,"completion_tokens":1411,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":750,"completion_tokens_details":{"reasoning_tokens":1324}},"tokens_in":750,"tokens_out":1411,"duration_ms":8828,"temperature":1.0,"reasoning_tokens":1324,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:36:30.329946+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Computing $(-1)^{|\\mu|-\\ell(\\mu)}\\langle\\nabla^r m_\\mu,s_\\lambda\\rangle$ for a concrete case such as $\\mu=\\lambda=(3,3), r=1$ via the recursion of Theorem 4.2 and the shuffle theorem and checking that every monomial coefficient is a nonnegative integer would settle the claim, since any negative coefficient would refute it (and the paper's computational check up to $|\\mu|=12$ finds none).","supporting_citations":[{"cited_title":"Bergeron, A","cited_arxiv_id":null,"evidence_quote":"formulated the conjecture and proved the integrality of the signed inner product for $r=1$."},{"cited_title":"Qu and G","cited_arxiv_id":null,"evidence_quote":"proved the compositional $(km,kn)$-shuffle theorem that expresses $\\nabla^r C_\\alpha(1)$ as a nonnegative sum of column LLT polynomials."},{"cited_title":"Haglund, Theq, t-Catalan numbers and the space of diagonal harmonics, University Lecture Series, vol","cited_arxiv_id":null,"evidence_quote":"established Schur positivity of LLT polynomials, converting the LLT sum into Schur positivity."},{"cited_title":"Blasiak, M","cited_arxiv_id":null,"evidence_quote":"gave an LLT expansion of $\\nabla^r s_\\lambda$ with integral coefficients, a key ingredient in the integrality statement (1) for all $r$."},{"cited_title":"Eğecioğlu and J","cited_arxiv_id":null,"evidence_quote":"proved integrality of the inverse Kostka matrix, used with [8] to obtain the integrality statement (1)."},{"cited_title":"Haglund, J","cited_arxiv_id":null,"evidence_quote":"introduced the $C_a$ operators and the basic identities (5)–(7) on which the entire recursion rests."},{"cited_title":"Eğecioğlu and J","cited_arxiv_id":null,"evidence_quote":"proved $e$-positivity of column LLT polynomials after $q\\mapsto q+1$, yielding Theorem 1.3."},{"cited_title":"Eğecioğlu and J","cited_arxiv_id":null,"evidence_quote":"gave the brick-tabloid transition formula used in Lemma 3.2 for the augmented monomial functions."},{"cited_title":"Carlsson and A","cited_arxiv_id":null,"evidence_quote":"proved the compositional shuffle theorem, the $r=1$ case that sets the pattern for the higher-power shuffle input."}],"review_version":2}