{"id":"3505329a-d480-41eb-9972-8b2312dd06e8","arxiv_id":"2412.20615","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For vexillary permutations, the double beta-Edelman-Greene coefficients are shown to be sums of Graham-positive monomials indexed by semistandard set-valued tableaux, with new restrictions on the types of factors appearing.","lead":"This paper gives an explicit tableau formula for vexillary double Edelman-Greene coefficients, proving a finer form of Graham positivity than the geometric positivity known before. The formula is fully combinatorial and manifestly positive, giving a new computational handle on Schubert calculus in K-theory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.5, the imported equality \\(\\leftarrow G_w = G^{\\phi(w)}_{\\lambda(w)}\\) from [12, Thm 5.8], is the load-bearing bridge; if that source covers only \\(\\beta=0\\) or only \\(\\beta=-1\\) rather than the formal-\\beta backstable setting, the tableau model in §3 does not compute the claimed…","rationale":"The proof architecture is otherwise largely self-contained: Lemma 3.2, Corollary 3.3, Proposition 3.5, and the type analysis in §4 are internally coherent, and the multiplicity bounds follow from the distinctness of cell weights noted in Remark 2.1. The central vulnerability is that the entire reduction to tableaux is one imported equality away from the object of Theorem 1.1. If Proposition 2.5 fails, neither the positivity nor the finer Type-1/Type-2 restriction would be about the double \\(\\beta\\)-Edelman–Greene coefficients \\(j^w_\\mu\\). The proof given for Proposition 2.5 is only a citation plus a shift argument; it does not spell out why [12, Thm 5.8] applies to the formal \\(\\beta\\) K-theoretic double Grothendieck polynomial with the present normalizations, nor does it justify the backstable limit in the presence of denominators \\(1+\\beta y_j\\). This is a verifiable gap rather than a suspected contradiction, so the reader's CONDITIONAL verdict is appropriate.","tokens_in":18891,"tokens_out":42461,"duration_ms":422886,"concrete_test":"Read the statement of [12, Thm 5.8] and check whether it is proved for the formal-\\beta double Grothendieck polynomial used in this paper, or only for \\(\\beta=0\\) (or \\(\\beta=-1\\)). Then independently verify Proposition 2.5 on a small vexillary \\(w\\) with a nontrivial flag spanning both signs, e.g. \\(w=s_1s_0\\) or a finite vexillary permutation shifted so that its flag contains both positive and negative entries, by computing \\(\\leftarrow G_w(\\beta;x;y)\\) from the limit \\(\\lim_{p\\to\\infty}\\gamma^{-p}G_{\\iota^p(w)}(\\beta;x_+;y_+)\\) to fixed degree and comparing coefficients with \\(G^{\\phi(w)}_{\\lambda(w)}(\\beta;x;y)\\).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 is proved by first replacing a vexillary \\(w\\in S_{\\mathbb{Z}}\\) with the flagged function \\(G^{\\phi(w)}_{\\lambda(w)}\\) via Proposition 2.5, then using (9) to identify \\(j^w_\\mu\\) with the coefficient \\(j^{\\lambda,\\phi}_\\mu\\). All of §3–§4, including the decomposition in Proposition 3.5 and the final \\(\\beta\\)-Graham-positive monomials, operates on \\(j^{\\lambda,\\phi}_\\mu\\). If Proposition 2.5 is not established for formal \\(\\beta\\) and for the backstable infinite setting, then the objects whose positivity is proved are not the double \\(\\beta\\)-Edelman–Greene coefficients of Theorem 1.1. The proof of Proposition 2.5 is a citation to [12, Thm 5.8] plus a shift-by-\\(\\iota\\) argument, so the cited theorem must cover the full double \\(\\beta\\)-Grothendieck polynomial \\(G_w(\\beta;x_+;y_+)\\), not only the \\(\\beta=0\\) double Schubert polynomial or only the \\(\\beta=-1\\) K-theory specialization, and must use the same flag and shape conventions. The passage from finite \\(S_\\infty\\) to \\(S_{\\mathbb{Z}}\\) by the limit defining \\(\\leftarrow G_w\\) also needs flag entries shifted by \\(p\\) and then shifted back by \\(\\gamma^{-p}\\); this is plausible but not shown in detail. A failure here would not be repaired by the later tableau arguments, since those arguments never refer back to \\(w\\) except through \\(\\lambda(w)\\) and \\(\\phi(w)\\).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a tableau formula for vexillary double β–Edelman–Greene coefficients j^w_µ(β;y), refining Anderson's β-Graham positivity theorem. The authors reduce the problem via Proposition 2.5 to flagged double stable β-Grothendieck functions, decompose flagged set-valued tableaux according to positive and non-positive entries, apply a Bender-Knuth symmetry argument to separate above- and below-diagonal parts, and then combine the two pieces using the ω involution. The main results, Theorem 1.1 and Theorem 4.7, assert that after multiplying by β^{ℓ(w)-|µ|}, each coefficient is a sum of β-Graham positive monomials, with a finer control on the types of terms that can occur: Type 3 terms appear at most twice per monomial, Type 1 or Type 2 at most once, and no monomial mixes Type 1 with Type 2.","tokens_in":19226,"tokens_out":24774,"duration_ms":237134,"significance":"If correct, this is a substantial combinatorial advance: it provides the first manifestly Graham-positive tableau description of double Edelman-Greene coefficients in the vexillary case, including the β=0 specialization, and it refines a geometric positivity result of Anderson. The proof is largely self-contained once the imported equality of Proposition 2.5 is granted, and it combines several interesting ingredients: a sign-based decomposition of set-valued tableaux, a Bender-Knuth symmetry for flagged skew functions, and an ω-involution step. The manuscript also includes useful appendix material establishing that backstable double β-Grothendieck polynomials expand into double stable β-Grothendieck functions. The main caveats are that Proposition 2.5 depends crucially on a cited theorem whose exact scope is not made explicit, and that two key lemmas (3.10 and 3.14) are compressed and contain apparent indexing errors. These issues are local and likely fixable, but they are load-bearing for the main theorem.","major_comments":[{"comment":"The entire reduction to tableaux hinges on the assertion that for vexillary w in S_Z, the backstable double β-Grothendieck polynomial equals the flagged double stable β-Grothendieck function. The proof says this is 'a straightforward consequence of [12, Thm 5.8]' and refers to a 'double Schubert polynomial equality' G_w(β;x_+;y_+) = G^{φ(w)}_{λ(w)}(β;x_+;y_+). Since Theorem 1.1 concerns formal β and the backstable infinite setting, the cited theorem must cover the β-Grothendieck polynomial defined in (8), not only the β=0 double Schubert polynomial or the β=-1 K-theory specialization, and it must use the same flag and shape conventions. Please state precisely what [12, Thm 5.8] proves, verify that it applies to the β-deformed functions in this paper, and spell out the γ^{-p} shift argument for S_Z. If [12] only treats β=0, then (9) is unsupported and the tableau model in Section 3 does not compute the double β-Edelman-Greene coefficients of Theorem 1.1.","section":"§2.3, Proposition 2.5"},{"comment":"Step 2 of Lemma 3.14 contains an indexing error that affects the proof of Lemma 3.7. Since χ(i) and χ(i+1) differ at row i+1, the violating eigenvalue bound should be ψ_{i+1} < k ≤ φ_{i+1}, not ψ_{i+1} < k ≤ φ_i; the text currently writes φ_i. Likewise, when ψ_{i+1} ≠ φ_{i+1}, the displayed identity should be ψ_{i+1} = i+1 - λ_{i+1} (from the definition ψ_i = min(i-λ_i, φ_i)), not ψ_{i+1} = i - λ_{i+1}. The subsequent statement that π_{i+1} sends j to j - ψ_{i+1} = j - (i+1) + λ_{i+1} for ψ_{i+1} < j ≤ φ_i also needs the upper bound corrected to φ_{i+1}. As written, the proof does not establish the equality (16) for all values in the relevant range. Please correct and reprove this step.","section":"§3.2, Lemma 3.14"},{"comment":"The Bender-Knuth argument proving symmetry in xi,...,xj is too compressed to be fully verifiable. The proof asserts a partition of tableaux into classes A_k with frozen entries and row segments, but it does not define the involution on free entries explicitly or check that the flag constraints and semistandard conditions are preserved when i and i+1 are swapped. Since Lemma 3.10 is used in Lemma 3.14 Step 1 to justify the symmetry that allows replacing π_i by π_{i+1}, this is a load-bearing step. Please expand the proof, ideally by giving the precise involution and verifying that it respects SetSSYT^φ_+(λ/µ).","section":"§3.2, Lemma 3.10"}],"minor_comments":[{"comment":"The abstract contains a small grammatical error: 'The goal of this paper to understand' should read 'The goal of this paper is to understand'.","section":"Abstract"},{"comment":"The phrase 'double Schubert polynomial equality' is inconsistent with the β-deformed setting; this should be 'double β-Grothendieck polynomial equality' or the scope should be clarified.","section":"§2.3, Proposition 2.5 proof"},{"comment":"The claim that w_{λ,φ} is 'the unique vexillary permutation with shape λ and flag φ' should be justified or given a reference, since Proposition 2.3 only guarantees existence up to flag equivalence.","section":"§5.1"},{"comment":"The sentence 'It would be an accomplishment to recover equivariant Grassmannian cohomology [13] or K-theory structure coefficients [21] in this way' is a bit vague; consider being more specific about what a recovery would mean, since those coefficients already have combinatorial rules.","section":"§5.5"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct after revision, but the main risk is the imported equality Proposition 2.5. I would ask the authors to confirm that [12, Thm 5.8] indeed covers the formal-β double Grothendieck polynomial and the backstable S_Z setting, and to fix the indexing in Lemma 3.14. If those points are resolved, the result is a nice contribution to the area."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper proves the first manifestly Graham-positive tableau formula for vexillary double Edelman–Greene coefficients, and the β=0 specialization is already new. I think the main theorem is correct, but there is one bridge that needs careful checking before I'd call it airtight.\n\nThe new content is real. Theorem 1.1, strengthened as Theorem 4.7, says that for a vexillary permutation, each coefficient expands into β-Graham-positive monomials with bounded multiplicity of Type 1/2/3 terms. Anderson's geometric positivity didn't give this finer information, and earlier combinatorial interpretations were not Graham positive even at β=0. The proof introduces a useful tool: the ψ flag and the permutation π that reindexes x-variables to make the below-diagonal contributions factor into Type 3 terms. The decomposition of tableaux into positive and non-positive parts is clean, and the reduction to the two special cases (φ non-negative / non-positive) via the involution is well organized.\n\nThe soft spots are not fatal, but they are real. The load-bearing step is Proposition 2.5, which identifies the backstable double β-Grothendieck polynomial of a vexillary permutation with the flagged double β-Grothendieck function. This is imported from [12, Thm 5.8], and that source is about double Schubert polynomials, i.e. β=0. The paper's shift-by-ι argument is plausible, but the formal-β version needs verification. If that equality fails, the rest of the paper computes a different object. I would ask the authors to either give a direct proof for formal β or cite a theorem that covers the needed generality. The other compressed spots, Lemma 3.10's Bender-Knuth argument and Lemma 3.14 Step 2, look repairable; I didn't find a break in the logic. No circular reasoning, no fitted parameters, citations are appropriate.\n\nWho should read this: algebraic combinatorics folks working on Schubert calculus, especially equivariant or K-theoretic positivity. It deserves a serious referee. I'd recommend sending it out and asking for an expanded treatment of Proposition 2.5. If that bridge holds, the paper is a solid advance.","headline":"Genuinely new refined Graham positivity for vexillary double Edelman–Greene coefficients; proof is plausible but requires verifying the imported formal-β flagged equality.","tokens_in":19745,"tokens_out":2794,"would_cite":true,"duration_ms":25533,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10","14M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Vexillary double Edelman–Greene coefficients admit a manifestly positive tableau expansion.","keywords":["vexillary permutations","double Edelman-Greene coefficients","beta-Graham positivity","flagged set-valued tableaux","backstable Grothendieck polynomials","double beta-Stanley symmetric functions","K-theory Schubert calculus"],"falsifier":"For the vexillary permutation w=345162, with λ(w)=(2,2,2,1) and φ(w)=(3,3,3,5), compute j^w_(2,2)(β;y) both by direct expansion of the backstable double β-Grothendieck polynomial and by the paper's tableau formula; a monomial containing both a Type 1 and a Type 2 factor, or any Type 3 factor more than twice, would disprove Theorem 1.1. The same comparison at β=0 against the double Schur expansion of the backstable Schubert polynomial would test the specialized corollary.","tokens_in":18667,"feed_emoji":"📐","tokens_out":15057,"duration_ms":134570,"temperature":0.7,"pith_summary":"For a vexillary permutation $w\\in S_{\\mathbb{Z}}$, the paper proves that the double $\\beta$-Edelman--Greene coefficient $j^w_\\mu(\\beta;y)$, multiplied by $\\beta^{\\ell(w)-|\\mu|}$, is a sum of monomials indexed by flagged set-valued tableaux. Every monomial is a product of factors $\\beta(y_i\\ominus y_j)$ with positive coefficient, so the expansion is manifestly $\\beta$-Graham positive. The formula is finer than previous positivity statements: in each monomial, a factor with $0<i<j$ or $i<j\\le 0$ appears at most once, a factor with $j\\le 0<i$ appears at most twice, and no monomial mixes the two one-sided factor types. Setting $\\beta=0$ gives the first manifestly Graham-positive tableau description of ordinary double Edelman--Greene coefficients. The result matters because these coefficients carry geometric information in equivariant cohomology and $K$-theory, where positive combinatorial rules are scarce.","feed_headline":"Tableau rule confirms finer positivity for vexillary EG coefficients","feed_subtitle":"A manifestly positive tableau rule for the vexillary coefficients, with Type 3 factors appearing at most twice.","key_machinery":"The central object is a flagged set-valued semistandard tableau (an $S$-tableau) with monomial weight $\\beta(x_i\\ominus y_j)$ per entry, and the load-bearing mechanism is the split of such a tableau into a non-positive part $T^-$ and a positive part $T^+$. Lemma 3.2 makes this split a bijection between $\\operatorname{SetSSYT}^{\\varphi}(\\lambda)$ and a disjoint union of products $\\operatorname{SetSSYT}^{\\varphi^-}(\\nu)\\times \\operatorname{SetSSYT}^{\\varphi^+}(\\lambda/\\mu)$ with disconnected skew piece, and Corollary 3.3 converts it into the convolution formula $j^{\\lambda,\\varphi}_\\rho=\\sum_\\nu j^{\\nu,\\varphi^-}_\\rho\\, j^{\\lambda,\\varphi^+}_\\nu$. Positivity of each factor is established by separating cells above and below the diagonal, by a custom permutation $\\pi$ of the $x$-variables (Lemma 3.7), and by an extension $\\tilde\\omega$ of the $\\omega$ involution that swaps non-positive and non-negative flags.","core_discovery":"The paper's central claim is Theorem 1.1, sharpened as Theorem 4.7: for every vexillary $w\\in S_{\\mathbb{Z}}$ and every partition $\\mu$, $\\beta^{\\ell(w)-|\\mu|} j^w_\\mu(\\beta;y)$ is a sum of monomials $\\prod \\beta(y_i\\ominus y_j)$, with each monomial containing at most one of the two one-sided factor types (Type 1: $0<i<j$; Type 2: $i<j\\le 0$) and at most two copies of any Type 3 factor ($j\\le 0<i$). The proof is combinatorial: it reduces the backstable double $\\beta$-Grothendieck polynomial of a vexillary permutation to the flagged double stable $\\beta$-Grothendieck function $G^{\\varphi(w)}_{\\lambda(w)}(\\beta;x;y)$, decomposes the set-valued tableaux into positive and non-positive parts, and analyzes the resulting skew shapes above and below the diagonal. The $\\beta=0$ specialization yields the first manifestly Graham-positive tableau rule for the double Edelman--Greene coefficients of vexillary permutations.","pith_inferences":["As an extension beyond the paper, if the type-restricted positivity is preserved under the transition equations the authors sketch, the theorem would propagate beyond vexillary permutations; the introduction's non-vexillary example shows the full Type 1/Type 2 separation cannot hold universally.","The constructive permutation $\\pi$ in Lemma 3.7 can be implemented as an algorithm that outputs the monomials directly, so one could compare its output term-by-term with the nonconstructive geometric positivity witness.","The same positive/non-positive tableau split may yield a tableau rule for equivariant $K$-theory Grassmannian structure coefficients, since the paper's final remarks connect double $\\beta$-Edelman--Greene coefficients to equivariant $K$-homology.","A direct empirical check is whether the 'at most twice' bound on Type 3 factors is ever attained; the paper does not exhibit an example of a monomial with a repeated Type 3 factor, so finding one would show the bound is sharp."],"forward_implications":["Every vexillary double $\\beta$-Edelman--Greene coefficient is a positive integer combination of $\\beta$-Graham monomials, so its expansion has no cancellation.","The refined type restriction is new: no monomial contains both Type 1 and Type 2 factors, Type 1 or Type 2 factors appear at most once, and Type 3 factors appear at most twice.","At $\\beta=0$, the formula gives the first manifestly Graham-positive tableau rule for double Edelman--Greene coefficients, including the vexillary double Schur coefficients.","The formula reduces nonvanishing of $j^w_\\mu$ to a polynomial-time check when $\\mu\\subseteq\\lambda(w)$ and $\\lambda(w)/\\mu$ has no diagonal cells, as the paper notes in Section 5.3.","The convolution formula factors any vexillary coefficient as $j^{\\lambda,\\varphi}_\\rho=j^{\\nu,\\varphi^-}_\\rho\\, j^{\\lambda,\\varphi^+}_\\nu$ for the unique $\\nu$ selected by Corollary 4.3."],"supporting_citations":[{"why":"Supplies the geometric β-Graham positivity theorem for double β–Edelman–Greene coefficients that the paper refines.","marker":"[1]"},{"why":"Gives the equality of vexillary double Grothendieck polynomials with flagged double Grothendieck functions, the key reduction behind Proposition 2.5.","marker":"[12, Thm 5.8]"},{"why":"Introduces backstable double Grothendieck polynomials, double β–Stanley symmetric functions, and the positivity conjecture, and supplies the divided-difference and ω-involution facts used here.","marker":"[14]"},{"why":"Provides the compatibility criterion for partitions and flags, used to identify λ(w), φ(w) and to handle negative flags via the involution.","marker":"[17, Prop 2.3]"},{"why":"Establishes the shape/flag data attached to vexillary permutations and the correspondence between vexillary permutations and compatible pairs.","marker":"[24]"}],"fun_headline_variants":["Bounded β-Graham positivity via tableau rule for vexillary EG","Tableau rule proves at most two Type-3 factors for vexillary EG","Manifestly positive tableau rule for vexillary double EG coefficients","Vexillary EG: explicit positive tableau formula with bounded factors","Tableau rule sharpens positivity: at most two Type-3 factors for vexillary EG"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument rests on the imported equality (Proposition 2.5, from [12, Theorem 5.8]) saying that for a vexillary permutation the backstable double β-Grothendieck polynomial equals the flagged double β-Grothendieck function; if that equality fails in the backstable infinite setting, the tableau formula would not compute the intended coefficients.","fun_headline_variants_meta":{"raw":{"variants":["Bounded β-Graham positivity via tableau rule for vexillary EG","Tableau rule proves at most two Type-3 factors for vexillary EG","Manifestly positive tableau rule for vexillary double EG coefficients","Vexillary EG: explicit positive tableau formula with bounded factors","Tableau rule sharpens positivity: at most two Type-3 factors for vexillary EG"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001237,"raw_usage":{"total_tokens":5058,"prompt_tokens":907,"completion_tokens":4151,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":4055}},"tokens_in":523,"tokens_out":4151,"duration_ms":29571,"temperature":1.0,"reasoning_tokens":4055,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:16:47.279790+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the vexillary permutation w=345162, with λ(w)=(2,2,2,1) and φ(w)=(3,3,3,5), compute j^w_(2,2)(β;y) both by direct expansion of the backstable double β-Grothendieck polynomial and by the paper's tableau formula; a monomial containing both a Type 1 and a Type 2 factor, or any Type 3 factor more than twice, would disprove Theorem 1.1. The same comparison at β=0 against the double Schur expansion of the backstable Schubert polynomial would test the specialized corollary.","supporting_citations":[{"cited_title":"Back stable K -theory Schubert calculus","cited_arxiv_id":null,"evidence_quote":"Introduces backstable double Grothendieck polynomials, double β–Stanley symmetric functions, and the positivity conjecture, and supplies the divided-difference and ω-involution facts used here."},{"cited_title":"Flagged Schur functions, Schubert polynomia ls, and symmetrizing operators","cited_arxiv_id":null,"evidence_quote":"Establishes the shape/flag data attached to vexillary permutations and the correspondence between vexillary permutations and compatible pairs."}],"review_version":1}