{"id":"e54c679e-020d-43ea-b618-0c6fb61ec2ba","arxiv_id":"2506.09421","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves Samuel's conjecture that triple Schubert calculus coefficients lie in the semiring N[t_i - y_j], and derives Kirillov's conjecture on the positivity of skew divided difference operators.","lead":"This paper proves a 2024 conjecture about the positivity of coefficients in products of double Schubert polynomials, and as a consequence settles a 2007 conjecture about skew divided difference operators. The proof works by refining a classical positivity theorem in equivariant cohomology and applying it to a geometric intersection in the flag variety.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of refined Graham positivity (Theorem 2.3) hinges on an unverified application of [1, Prop. 19.4.4] to the pair B^-(w s_i) ⊂ B^-(w); if those hypotheses fail, Theorem 1.1 and Corollary 1.2 collapse.","rationale":"I read the paper as proving Samuel's and Kirillov's conjectures through a refined Graham positivity theorem plus a geometric interpretation of triple Schubert coefficients. The geometric intersection in Lemma 2.5 appears plausible: the reduction to a Richardson variety via the permutation u_0 is standard, and my checks of small examples did not reveal an error there. The genuinely load-bearing gap is the induction step of Theorem 2.3: the proof delegates to Anderson–Fulton Proposition 19.4.4 a decomposition that is essential for the induction, but the hypotheses of that proposition are not demonstrated. This is not a stylistic concern; if the proposition is inapplicable, the refined positivity theorem lacks a proof. The reader's weakest_assumption identified exactly this delegation, and my independent reading agrees. I therefore see no reason to change the CONDITIONAL verdict: the paper is promising and likely correct, but a referee must verify the cited proposition's hypotheses before full acceptance.","tokens_in":5692,"tokens_out":61014,"duration_ms":586764,"concrete_test":"Obtain [1, Proposition 19.4.4] and its surrounding hypotheses. Verify explicitly for G = GL_n and any w s_i > w that the quotient B^-(w)/B^-(w s_i) is a one-dimensional unipotent group with T-weight -wα_i and that the required normal-bundle condition holds; then test the minimal case n = 2, w = id, i = 1 by constructing Z_1, Z_2 for Y = X_{s_1} and comparing the identity [Y] = [Z_1] + χ[Z_2] via localization. If the hypotheses fail or the identity does not hold, the induction step is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The induction in Section 2.1 is the keystone of the paper: Theorem 2.3 is established only by asserting, with a bare page reference to [1, p. 384], that the subgroup pair B^-(w s_i) ⊂ B^-(w) satisfies the hypotheses of [1, Proposition 19.4.4] with character χ = -wα_i. Lemma 2.2 confirms normality of B^-(w s_i) in B^-(w), but Anderson–Fulton's proposition has additional hypotheses that are not checked in the text. Specifically, the proposition must produce B^-(w)-invariant effective cycles Z_1, Z_2 realizing [Y] = [Z_1] + χ[Z_2]; without this decomposition the induction over ℓ(w) cannot proceed. Typical required conditions include that the quotient B^-(w)/B^-(w s_i) is a one-dimensional unipotent group with a specified T-weight, and that the T-action on normal spaces satisfies a linearity/normal-bundle condition. None of these are verified here. If the hypotheses of [1, Prop. 19.4.4] fail, the refined Graham positivity theorem—and hence Theorem 1.1 and its corollary—would lack proof. A referee must confirm the applicability of this delegation before the central claim can be accepted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Samuel's conjecture on Graham positivity of the expansion coefficients of products of double Schubert polynomials in three sets of variables (Theorem 1.1), and as a corollary Kirillov's conjecture on positivity of skew divided difference operators applied to Schubert polynomials (Corollary 1.2). The proof introduces a refined Graham positivity theorem (Theorem 2.3) for B^-(w)-invariant effective cycles and gives a geometric interpretation of the coefficients as an intersection of translated Schubert varieties (Lemma 2.5, proof of Theorem 1.1). The paper is concise and delegates substantial technical steps to Anderson and Fulton's book [1].","tokens_in":5947,"tokens_out":12112,"duration_ms":128666,"significance":"If the proof is completed, the paper resolves two previously open conjectures and establishes a refined positivity theorem that strengthens Graham's theorem. The geometric explanation of Billey's formula is a nice byproduct, and the K-theoretic conjecture at the end suggests further work. The overall strategy is plausible and the paper is clearly written, but two load-bearing technical steps—the application of [1, Prop. 19.4.4] and the proof of Lemma 2.5—are not adequately justified and must be repaired before the claims can be accepted.","major_comments":[{"comment":"The induction step asserts that the pair B^-(w s_i) ⊂ B^-(w) satisfies the hypotheses of [1, Proposition 19.4.4] with character χ = -w α_i, citing only [1, p. 384]. Lemma 2.2 establishes only that N^-(w s_i) is a normal subgroup of N^-(w). The cited proposition likely requires additional conditions (for instance, that the quotient is a one-parameter unipotent group with a specified T-action and that a normal-bundle condition holds); none of these are verified in the text. Because this proposition is the only tool that produces the decomposition [Y]_T = [Z_1]_T + χ[Z_2]_T, the induction is not established. The authors must state the hypotheses of [1, Prop. 19.4.4] explicitly and verify them for the pair B^-(w s_i) ⊂ B^-(w).","section":"Section 2.1, proof of Theorem 2.3"},{"comment":"The proof of Lemma 2.5 is incorrect as written. The claim 'Since v∈S_n, s_i v > v for n≤i<2n, and thus B^-vB/B is invariant under s_i' is false: left multiplication by s_i sends B^-vB/B to B^- s_i v B/B, a different Schubert cell, and right multiplication by s_i sends it to B^- v s_i B/B, also different because v s_i ≠ v. Consequently the conclusion B^-vB/B = u_0 B^-vB/B is unjustified. This lemma is load-bearing for the geometric formula in the proof of Theorem 1.1; the authors need to supply a valid proof of properness and generic transversality of the intersection, or give a precise reference with a clear explanation of how it applies.","section":"Section 2.3, Lemma 2.5"}],"minor_comments":[{"comment":"The reference to 'Theorem 2.2' should be 'Lemma 2.2'.","section":"Section 2.1, proof of Theorem 2.3"},{"comment":"The reference to 'Theorem 2.5' should be 'Lemma 2.5'.","section":"Section 2.3, proof of Theorem 1.1"},{"comment":"After Lemma 2.5, the equality [τ B^-uB/B ∩ B^-vB/B]_T = Σ_w c^w_{u,v}(y,t) [B^-wB/B]_T is asserted in one sentence; the authors should explicitly state that the intersection cycle represents the equivariant product of the two classes and that the coefficients match the polynomial expansion (1).","section":"Section 2.3, proof of Theorem 1.1"},{"comment":"The phrase 'finite many' should be 'finitely many'.","section":"Section 2.1, Corollary 2.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is very short and relies heavily on Anderson–Fulton [1] for two critical steps. The gaps identified in the major comments are substantial but likely fixable; the authors should be encouraged to expand the proof with full details. The subject fits the journal's scope. The resolution of Samuel's and Kirillov's conjectures would be a valuable contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves Samuel's conjecture on triple Schubert coefficients and, as a corollary, Kirillov's conjecture on skew divided difference operators. That is the real news, and it is likely correct. The novelty is a refined Graham positivity theorem (Theorem 2.3), which constrains the roots to the inversion set of w for a B^-(w)-invariant cycle. This is strictly stronger than Graham's classical statement and it does real work: it gives a geometric explanation of Billey's formula and a clean proof of the main theorem via an intersection interpretation of c^w_{u,v}(y,t) in a doubled flag variety. Section 2.4's conjecture for Grothendieck polynomials is a natural add-on, not central.\n\nWhat I like: the geometric setup in Section 2.3 is neat, and the proof of Theorem 1.1 is genuinely short once the refined positivity is granted. The paper is honest about relying on Anderson-Fulton for the heavy lifting.\n\nThe soft spot is the delegation. Theorem 2.3 is proved by induction, and the induction step is an application of [1, Prop. 19.4.4] to the pair B^-(ws_i) ⊂ B^-(w) with χ = -wα_i. The hypotheses of that proposition are not stated, and the authors assert 'the pair satisfies the condition' with a bare page reference. That is a checkable claim, but a referee must check it, because the proposition is the keystone. There is also a likely typo: the text says the cycles Z_1, Z_2 are B^-(w)-invariant. For the induction to work they need to be B^-(ws_i)-invariant; otherwise the extra factor χ will not stay in the desired semiring. I assume this is a typo, but as written it is a real gap in the written proof. The proof of Lemma 2.5 is also delegated to [1, Section 19.3]; the sketch via the longest element of W and the commutation relation is plausible, but a referee should confirm properness and transversality.\n\nIf those two citations check out, the paper is solid and the conjectures are resolved. It is a serious paper for people in Schubert calculus and equivariant cohomology, and it deserves refereeing. My recommendation: send to a referee with explicit requests to verify the Anderson-Fulton hypotheses and to ask the authors to fix the invariance statement for Z_i.","headline":"Proves two conjectures via a plausible refined Graham positivity; main caveat is an unverified delegation to Anderson-Fulton.","tokens_in":6475,"tokens_out":7971,"would_cite":true,"duration_ms":83982,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","14M15","14N15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The coefficients expressing a product of two double Schubert polynomials in three sets of variables are shown to be nonnegative combinations of $t_i - y_j$.","keywords":["double Schubert polynomials","Graham positivity","equivariant cohomology","Schubert calculus","skew divided difference operators","flag varieties","Samuel's conjecture","Kirillov's conjecture"],"falsifier":"For a small triple such as $u=v=w=s_1$ in $S_3$, expand $S_{s_1}(x;y)S_{s_1}(x;t)$ in the basis $S_w(x;t)$ and check whether every coefficient $c^w_{u,v}(y,t)$ is a nonnegative integer polynomial in $t_i - y_j$; any negative coefficient would falsify Theorem 1.1.","tokens_in":5470,"feed_emoji":"➕","tokens_out":16716,"duration_ms":140970,"temperature":0.7,"pith_summary":"The paper proves Samuel's conjecture that, in the expansion of the product of two double Schubert polynomials $S_u(x;y)S_v(x;t)$ as a combination of single double Schubert polynomials $S_w(x;t)$, every coefficient $c^w_{u,v}(y,t)$ is a polynomial with nonnegative integer coefficients in the differences $t_i - y_j$. The proof works by establishing a refined version of Graham's positivity theorem: an effective cycle in a flag variety that is invariant under the subgroup $B^-(w)$ has an equivariant cohomology class expressible as a combination of effective cycle classes with coefficients built from the negatives of the inversion roots of $w$. This refined statement is then applied to a geometric realization of $c^w_{u,v}(y,t)$ as the equivariant class of an intersection of two Schubert varieties. As a direct corollary, the paper settles Kirillov's conjecture that skew divided difference operators applied to Schubert polynomials yield polynomials with nonnegative coefficients.","feed_headline":"Schubert product coefficients are Graham-positive","feed_subtitle":"Refined positivity theorem settles Samuel's and Kirillov's conjectures in one stroke.","key_machinery":"The load-bearing object is the refined Graham positivity theorem (Theorem 2.3), which asserts that $B^-(w)$-invariant effective cycles have equivariant classes lying in the cone generated by the negatives of the inversion roots of $w$: $[Y]_T \\in \\sum_i \\mathbb{N}[-\\alpha]_{\\alpha \\in I(w)} \\cdot [Z_i]_T$. The proof is an induction on the length of $w$; the inductive step uses Anderson–Fulton's Proposition 19.4.4 on the subgroup pair $B^-(w s_i) \\subset B^-(w)$ with character $\\chi = -w\\alpha_i$ to split the class of a $B^-(w s_i)$-invariant cycle into effective pieces. The machinery also includes a new geometric interpretation (Section 2.3) of $c^w_{u,v}(y,t)$ as the equivariant class of the intersection $\\tau B^-uB/B \\cap B^-vB/B$ expanded in the Schubert basis, where the special permutation $\\tau$ has inversion set $\\{y_j - t_i\\}$; this is what converts the abstract positivity into the concrete statement in $t_i - y_j$.","core_discovery":"On the paper's own terms, the central discovery is that the refined Graham positivity theorem (Theorem 2.3) holds: for a $B^-(w)$-invariant effective cycle $Y$ on a non-singular variety $X$ with a $B^-$ action, $[Y]_T$ lies in the semiring generated by the negative inversion roots of $w$, with $B^-$-invariant effective cycles as coefficients: $[Y]_T \\in \\sum_i \\mathbb{N}[-\\alpha]_{\\alpha \\in I(w)} \\cdot [Z_i]_T$ in $H^*_T(X)$. This strengthens Graham's original positivity theorem, which allows all positive roots, by restricting the cone of allowed characters to those of the form $-\\alpha$ for $\\alpha \\in I(w)$. Applying this to the intersection $\\tau B^-uB/B \\cap B^-vB/B$ realizes the coefficients $c^w_{u,v}(y,t)$ as the coefficients of this class in the Schubert basis, and the inversion set of $\\tau$ is exactly $\\{y_j - t_i\\}$, giving the required positivity. Setting $y=0$ then yields Kirillov's conjecture.","pith_inferences":["The refined positivity theorem is stated for arbitrary reductive groups, so the same argument may yield analogues of Samuel's and Kirillov's conjectures in other Lie types once the appropriate equivariant classes are identified.","If the Anderson–Fulton condition is verified explicitly, the induction in Theorem 2.3 could be made algorithmic, potentially producing a positive combinatorial formula for $c^w_{u,v}(y,t)$ rather than a purity statement.","The conjecture for double Grothendieck polynomials (Conjecture 2.7) suggests that the positivity should survive $K$-theoretic deformation; a $K$-theoretic analogue of the refined Graham theorem would be the natural tool."],"forward_implications":["Samuel's conjecture holds: for every $u,v,w \\in S_\\infty$, the coefficient $c^w_{u,v}(y,t)$ is a nonnegative integer polynomial in the differences $t_i - y_j$.","Kirillov's conjecture holds: for every $u,v,w$, the skew divided difference operator $\\partial_{w/v}$ applied to $S_u(x)$ produces a polynomial with nonnegative coefficients.","The refined Graham positivity theorem (Theorem 2.3) is valid for any $B^-(w)$-invariant effective cycle, generalizing Graham's theorem and giving a geometric explanation of the positivity in Billey's formula.","The coefficients $c^w_{u,v}(y,t)$ are realized geometrically as the Schubert-expansion coefficients of the equivariant class of the intersection $\\tau B^-uB/B \\cap B^-vB/B$, which ties the positivity ring to the inversion set of $\\tau$."],"supporting_citations":[{"why":"Supplies Proposition 19.4.4 used in the induction step of the refined Graham positivity theorem, Section 19.3 for the proper/transverse intersection claim, and the identification of double Schubert polynomials with equivariant classes.","marker":"[1]"},{"why":"Graham's positivity theorem that Theorem 2.3 refines; provides the original result and the baseline positivity statement generalized here.","marker":"[8]"},{"why":"States Samuel's conjecture that Theorem 1.1 proves; also gives the identity $\\partial_{w/v}S_u(x;y) = c^w_{u,v}(y,x)$ used to derive Corollary 1.2.","marker":"[21]"},{"why":"States Kirillov's conjecture that Corollary 1.2 proves, and gives the definition of skew divided difference operators.","marker":"[11]"},{"why":"Provides an alternative derivation of the identity $\\partial_{w/v}S_u(x;y) = c^w_{u,v}(y,x)$, used in the proof of Corollary 1.2.","marker":"[6]"}],"fun_headline_variants":["Refined Graham positivity settles Samuel and Kirillov conjectures","Graham positivity from negative inversion roots proves two conjectures","Single refined theorem: Graham positivity for triple Schubert calculus","Triple Schubert positivity: negative roots drive Graham expansion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the subgroup pair $B^-(w s_i) \\subset B^-(w)$ satisfies the hypotheses of Anderson–Fulton's Proposition 19.4.4 with character $\\chi = -w\\alpha_i$, a condition the text asserts with a page reference but does not verify.","fun_headline_variants_meta":{"raw":{"variants":["Refined Graham positivity settles Samuel and Kirillov conjectures","Graham positivity from negative inversion roots proves two conjectures","Single refined theorem: Graham positivity for triple Schubert calculus","Triple Schubert positivity: negative roots drive Graham expansion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000492,"raw_usage":{"total_tokens":2353,"prompt_tokens":814,"completion_tokens":1539,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":430,"completion_tokens_details":{"reasoning_tokens":1471}},"tokens_in":430,"tokens_out":1539,"duration_ms":12993,"temperature":1.0,"reasoning_tokens":1471,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:50:56.136384+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a small triple such as $u=v=w=s_1$ in $S_3$, expand $S_{s_1}(x;y)S_{s_1}(x;t)$ in the basis $S_w(x;t)$ and check whether every coefficient $c^w_{u,v}(y,t)$ is a nonnegative integer polynomial in $t_i - y_j$; any negative coefficient would falsify Theorem 1.1.","supporting_citations":[{"cited_title":"Cambridge University Press, Cambridge, 2024","cited_arxiv_id":null,"evidence_quote":"Supplies Proposition 19.4.4 used in the induction step of the refined Graham positivity theorem, Section 19.3 for the proper/transverse intersection claim, and the identification of double Schubert polynomials with equivariant classes."},{"cited_title":"Positivity in equivariant Schubert calculus.Duke Math","cited_arxiv_id":null,"evidence_quote":"Graham's positivity theorem that Theorem 2.3 refines; provides the original result and the baseline positivity statement generalized here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States Samuel's conjecture that Theorem 1.1 proves; also gives the identity $\\partial_{w/v}S_u(x;y) = c^w_{u,v}(y,x)$ used to derive Corollary 1.2."},{"cited_title":"Kirillov","cited_arxiv_id":null,"evidence_quote":"States Kirillov's conjecture that Corollary 1.2 proves, and gives the definition of skew divided difference operators."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides an alternative derivation of the identity $\\partial_{w/v}S_u(x;y) = c^w_{u,v}(y,x)$, used in the proof of Corollary 1.2."}],"review_version":1}