{"id":"9344ca44-19db-4287-9f18-f7c7478acae1","arxiv_id":"2608.10516","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Skew Schur, Schur P, and Schur Q polynomials are realizable volume polynomials via Richardson varieties, settling the Huh-Matherne-Mészáros-St. Dizier Lorentzian conjectures.","lead":"The paper proves that the coefficients of skew Schur polynomials and their shifted analogues satisfy strong concavity because these polynomials are intersection volumes of Richardson varieties in Grassmannians. This settles two open Lorentzian conjectures and gives new inequalities for tableau counts.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Type C stable Schubert homomorphism (Prop. 5.3) is the load-bearing step; no error found, but an independent normalization check of (5.18) would settle it.","rationale":"I read the paper in good faith. The type A chain (Richardson–Pieri identity + total-Chern volume) is internally consistent and I found no gap. The type C chain is more delicate: Proposition 5.3 is a proof, not a citation, but its key Giambelli normalization comes from [19] and the paper's own remarks emphasize that a literal substitution would be wrong. I traced the small cases (straight λ=(1),(2),(3) and skew (3)/(1)) and the coefficients matched the intersection numbers, so I do not believe the concern actually lands. However, because the entire shifted volume theorem would collapse if a factor of 2^{ℓ(ν)} were missing, this is the single most load-bearing assumption; an independent computational check of (5.18) for a new shape is the right way to close it. Thus I do not change the reader's ACCEPT.","tokens_in":24080,"tokens_out":59522,"duration_ms":438034,"concrete_test":"Use a computer algebra system to verify (5.18) for a nontrivial skew case: take strict λ=(3,1), μ=(1), n=3, α=(1,1,1). Compute the left side [x y z]P_{(3,1)/(1)} via Stembridge's marked shifted tableau rule (or the Hopf recurrence of Lemma 5.1), and compute the right side ∫_{X_{(4,2)}∩X_{(1)}} c_1(E)^3 in the Chow ring of LG(4,8) using the Kresch–Tamvakis Giambelli formula. If the two integers differ, Proposition 5.3 is false as used; if they agree, repeat for a few additional skew shapes. The check directly isolates the normalization-sensitive step.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central type C claim that N(P_{λ/μ}) and N(Q_{λ/μ}) are realizable volume polynomials rests on the coefficient identity (5.18), which in turn depends on Proposition 5.3: the stable Schubert homomorphism φ_N: Γ_Z → A^*(LG(N,2N)) with q_r ↦ c_r(E) and Q_ν ↦ σ_ν for ν⊆ρ_N. This is the one step that the paper itself flags as normalization-sensitive (Section 1.2: 'the type C passage is not the false literal substitution Q_λ(E)=σ_λ'). The proof delegates the crucial Giambelli normalization to Kresch–Tamvakis [19]; if the modified Q representative were off by a factor depending on ℓ(ν) (or if the Pfaffian convention differed), every coefficient in (5.18) would acquire a spurious scalar, and Theorem 5.6 would realize a multiple of N(P_{λ/μ}) rather than the polynomial itself. Neither the proof nor Remark 5.5 (which checks one two-row case) excludes such a mismatch for general skew shapes. Since Theorems 1.1, 5.8, and all shifted inequalities in Sections 6-8 inherit this identification, this is the most load-bearing assumption in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the factorial normalizations N(s_{λ/μ}), N(P_{λ/μ}), and N(Q_{λ/μ}) are realizable volume polynomials over C for every ordinary skew shape and every strict skew shape, thereby settling Conjectures 19 and 20 of Huh–Matherne–Mészáros–St. Dizier and strengthening the Schur-P statement to arbitrary skew shifted shapes. The method identifies skew Schur and skew Schur P/Q-functions as top-degree total-Chern intersection polynomials on Richardson varieties in ordinary and Lagrangian Grassmannians, then invokes a general total-Chern volume theorem (Cid-Ruiz) and an operator-theoretic upgrade (Grund–Huh–Michałek–Süss–Wang) to pass from denormalized Lorentzianity to volume realizability. The paper further derives a substantial package of consequences: reverse Khovanskii–Teissier inequalities, Hessian-signature and principal-minor inequalities, root-direction log-concavity, dominance monotonicity, exact support permutahedra for ordinary skew Schur functions, straight Schur P/Q support polytopes, and log-concavity theorems for cumulative two-row Littlewood–Richardson coefficients.","tokens_in":24180,"tokens_out":55921,"duration_ms":430049,"significance":"If the main results stand, this is a significant advance in the theory of Lorentzian and volume polynomials. The geometric realization on Richardson varieties is natural, and the passage from Lorentzianity to volume realizability is a genuine strengthening. The type C construction via the stable Schubert homomorphism is a novel and useful ingredient, and the resulting inequalities for tableau multiplicities and Littlewood–Richardson coefficients are numerous and interesting. The paper is careful to attribute general mechanisms to prior work, and the central coefficient identities are derived explicitly and checked against small examples. The main volume-realization theorems are supported by coherent proofs; the principal concern identified in this report is a local error in one of the formal consequence theorems, not in the central argument.","major_comments":[{"comment":"The formula for Pol_μ(f) in Eq. (6.4) is not the full polarization, because its diagonal specialization does not recover f. For a concrete counterexample, take F = x_1^2 + x_1 x_2, so f = N(F) = x_1^2/2 + x_1 x_2 and μ = (2,1). Formula (6.4) yields Pol_μ(f) = (1/2)e_2(Y_1) + 2 e_1(Y_1)e_1(Y_2); specializing y_{i,k}=x_i gives (1/2)x_1^2 + 4x_1 x_2, not f. The correct normalization is division by ∏_i (μ_i choose α_i), not multiplication. This invalidates the proof that Pol_μ(f) is a realizable volume polynomial (the proof explicitly relies on the false diagonal claim) and consequently the use of this theorem in Corollary 6.9. The support statement (6.5) is unaffected, and the central volume-realization theorems do not depend on this formula, but the error must be corrected in a revision.","section":"§6.1, Eq. (6.4)"}],"minor_comments":[{"comment":"Since the paper itself flags the stable Schubert homomorphism as the normalization-sensitive step, it would be helpful to add a remark explicitly identifying the Kresch–Tamvakis substitution e_r for q_r with q_r ↦ c_r(E) in this setting, and to verify the normalization in a case with ℓ(ν)>1 (e.g., ν=(2,1)) in addition to the ℓ(ν)=1 check in Remark 5.5. I have not found an actual error here, but such a remark would address the residual uncertainty about a possible scalar mismatch.","section":"§5.1, Prop. 5.3"},{"comment":"The phrase 'has no internal zeros' should be clarified to mean that the positive entries form an interval without gaps. Taken literally, the assertion is false: for P_{(4,2)}(x,y) = x^4 y^2 + 2x^3 y^3 + x^2 y^4, the sequence B_0,...,B_6 is (0,0,1,2,1,0,0), which has zeros at r=1 and r=5. The intended interval property follows from M-convexity and is sufficient for the stated log-concavity.","section":"§8.1, Thm. 8.7"},{"comment":"The text repeats the claim that the expression in (6.4) is 'the usual full polarization' whose diagonal specialization is f. After correcting the normalization factor, the proof should be updated consistently, and it would be useful to state explicitly which convention from [13, Proposition 4.1] is being followed.","section":"§6.1, proof of Thm. 6.2(iii)"}],"recommendation":"major_revision","confidential_remarks":"The paper is strong and the central claims are likely correct. The main issue is the polarization formula in Theorem 6.2(iii), which is local and readily fixable; the rest of the paper's core results do not depend on it. I do not share the strongest form of the type C normalization concern, since the step is a direct citation of a published theorem and the local consistency checks are reassuring. After the authors correct Eq. (6.4) and re-verify the affected consequences, the paper should be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This one is worth a careful read. The paper proves that N(s_{λ/μ}), N(P_{ϑ}), and N(Q_{ϑ}) are realizable volume polynomials, settling Conjectures 19 and 20 of Huh–Matherne–Mészáros–St. Dizier and going beyond them to arbitrary skew P/Q-functions. The main new ingredients are a Richardson–Pieri identity in type A and a stable-Q Richardson identity in type C, packaged as cycle transforms on Richardson varieties. Those identities convert skew Schur and P/Q coefficients into total-Chern intersections, and then Cid-Ruiz's theorem plus the Grund–Huh–Michałek–Süss–Wang operator machinery upgrades Lorentzianity to volume realizability. The paper is transparent about which parts are new and which are recoveries: the ordinary skew and straight P/Q support results are already known, and it says so.\n\nThe soft spot is the one the author names: Proposition 5.3, the stable Schubert homomorphism sending q_r to c_r(E) and Q_ν to σ_ν. The whole type C argument rests on this normalization. If the modified ~Q_ν representative were off by a factor depending on ℓ(ν), every coefficient in (5.18) would acquire a spurious scalar and Theorem 5.6 would realize a multiple of N(P_{λ/μ}) rather than the polynomial itself. The proof delegates to Kresch–Tamvakis, and Remark 5.5 checks only a two-row case. I found no error, and the statement is standard in form, but an independent check of (5.18) against a three- or four-row example would settle it. That is my only substantive concern, and it is a referee-level check, not a fatal flaw.\n\nThe inequality package that follows — reverse Khovanskii–Teissier, Hessian sign, dominance monotonicity, two-row log-concavity — is broad and correctly derived from the volume realization. Some of these overlap with Chin–Qin's symmetric Lorentzian results; the paper acknowledges that. Minor point: for general skew P/Q shapes the support theorem gives an exact permutahedron but no closed formula for the dominant partition; again acknowledged.\n\nWho is this for? Anyone working on Lorentzian polynomials, Schur positivity, or Schubert calculus. It deserves a serious referee. I would send it to peer review, with a specific request to verify the type C normalization.","headline":"Strong paper resolving two open Lorentzian conjectures with a realizable-volume upgrade; the main risk is the type C Schubert normalization step, which the author flags and which deserves an independent check.","tokens_in":24830,"tokens_out":3005,"would_cite":true,"duration_ms":27729,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","14M15","14C17","52B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every ordinary skew Schur polynomial and every skew Schur P- or Q-function is, after factorial normalization, a realizable volume polynomial over the complex numbers.","keywords":["skew Schur polynomial","Schur P-function","Schur Q-function","Richardson variety","Lagrangian Grassmannian","total Chern class","realizable volume polynomial","Lorentzian polynomial"],"falsifier":"Take a small strict skew shape, for instance $\\lambda=(2,1)$, $\\mu=\\emptyset$, with $n=2$ and $\\alpha=(2,1)$. Compute $[x^\\alpha]P_{\\lambda/\\mu}$ by the marked shifted-tableau generating function and separately compute $\\int_{R^C_{\\lambda/\\mu}} c_2(E)c_1(E)$ on the Lagrangian Richardson variety; the paper's theorem says the two numbers are equal (both equal 1). Any mismatch in such a finite computation would falsify the type-C volume theorem, and the same check can be run for ordinary skew Schur polynomials with the type-A Richardson identity.","tokens_in":23739,"feed_emoji":"📐","tokens_out":16632,"duration_ms":132397,"temperature":0.7,"pith_summary":"The paper proves that, in any finite number of variables, the factorial normalization of every ordinary skew Schur polynomial and every skew Schur P- or Q-function is a realizable volume polynomial over the complex numbers. A realizable volume polynomial is the top self-intersection of a linear combination of semiample divisors on an irreducible projective variety, a strictly stronger property than being Lorentzian. The proof identifies these symmetric-function coefficient arrays with top-degree total-Chern intersection numbers on Richardson varieties in ordinary and Lagrangian Grassmannians. This settles the two open Lorentzian conjectures for skew Schur and Schur P functions, extends the P/Q statement to arbitrary strict skew shapes, and yields a package of coefficient inequalities including reverse Khovanskii–Teissier bounds, Hessian-signature and principal-minor conditions, root-direction log-concavity, and dominance monotonicity toward balanced contents.","feed_headline":"All normalized skew Schur and Schur P/Q functions are volume polynomials","feed_subtitle":"A Grassmannian intersection model proves log-concavity and settles the Lorentzian conjectures for every skew shape.","key_machinery":"The load-bearing object is the Richardson cycle transform: for an irreducible $d$-dimensional subvariety $X$ of an ordinary Grassmannian, $\\Theta^A_{X,n}(x) = \\sum_{\\nu \\subseteq c^r,\\, |\\nu|=d} (\\int_X s_\\nu(S^\\vee|_X))\\, s_\\nu(x)$, with the type-C analogue on a Lagrangian Grassmannian using Schubert classes $\\sigma_\\nu$ and Schur $P$-functions; the skew Schur and skew $P/Q$ cases are the specializations where $X$ is the corresponding Richardson variety. The transform converts tableau coefficients into top-degree total Chern intersections. The engine behind the volume conclusion is a general theorem: the top-degree total-Chern polynomial of globally generated bundles is denormalized Lorentzian, and its factorial normalization is a realizable volume polynomial, realized on a fiber product of projective bundles whose tautological divisors reproduce the Chern classes as intersection powers; a realizable-covolume differential operator removes the extra divisor powers. The type-C step uses a stable graded ring homomorphism from the Schur-$Q$ algebra to the Chow ring of the Lagrangian Grassmannian that sends $q_r$ to $c_r(E)$ and, through a modified Pfaffian representative, $Q_\\nu$ to the Schubert class $\\sigma_\\nu$; this matching is what makes the shifted $P/Q$ identity true.","core_discovery":"On its own terms, the central result is Theorem 1.1: for every ordinary skew shape $\\theta$, every strict skew shape $\\vartheta$, and every $n \\geq 1$, the polynomials $\\mathcal{N}(s_\\theta(x_1,\\ldots,x_n))$, $\\mathcal{N}(P_\\vartheta(x_1,\\ldots,x_n))$, and $\\mathcal{N}(Q_\\vartheta(x_1,\\ldots,x_n))$ are realizable volume polynomials over $\\mathbb{C}$, with smooth irreducible realizations whenever nonzero. The proof runs through two Richardson identities: $[x^\\alpha]s_{\\lambda/\\mu} = \\int_{R^A_{\\lambda/\\mu}} \\prod_i h_{\\alpha_i}(S^\\vee)$ in type A, and $[x^\\alpha]P_{\\lambda/\\mu} = \\int_{R^C_{\\lambda/\\mu}} \\prod_i c_{\\alpha_i}(E)$ in type C, where the second uses a stable homomorphism from the Schur-$Q$ algebra to the Chow ring of the Lagrangian Grassmannian sending a modified representative of $Q_\\nu$ to the Schubert class $\\sigma_\\nu$. These identities turn tableau counts into top-degree total-Chern intersections, and a general volume theorem upgrades the resulting polynomials to realizable volumes. The $Q$-function case follows because $Q_{\\lambda/\\mu} = 2^{\\ell(\\lambda)-\\ell(\\mu)}P_{\\lambda/\\mu}$.","pith_inferences":["A direct next step the paper leaves open is a closed shifted-tableau description of the dominant partition $\\kappa^P_{\\lambda/\\mu,n}$ that generates the skew P/Q support permutahedron; small examples from the P-expansion could be used to guess and test such a formula.","The type-C matching is normalization-sensitive, so analogous modified representatives would be needed before the same volume construction could be attempted on other isotropic flag varieties, such as odd orthogonal or other symplectic types.","A natural stress test of the general mechanism is to compute the cycle transform for a non-Richardson subvariety of a small Grassmannian and verify the volume inequalities on that example.","The covariance bound and ultra-log-concavity for factorially tilted content distributions are proved from the volume realization alone, so checking them on the mixed products in Corollary 8.8 would separate the volume mechanism from special tableau structure."],"forward_implications":["The two Lorentzian conjectures for skew Schur and for Schur P functions are true, and the P/Q statement holds for every strict skew shape, not only straight shapes.","Every finite-variable ordinary or shifted tableau coefficient array inherits the full Hodge–Riemann package: reverse Khovanskii–Teissier inequalities, one-positive-eigenvalue Hessians, alternating principal-minor signs, and log-concavity along coordinate root directions.","Supports are exactly lattice points of permutahedra: for ordinary skew Schur polynomials the support is $P_{\\kappa(\\lambda/\\mu)} \\cap \\mathbb{Z}^n$ with unit vertex coefficients; for straight Schur P/Q the classical supports are recovered, and skew P/Q admit an exact permutahedron support.","Cumulative two-row ordinary Littlewood–Richardson coefficients and weighted cumulative two-row shifted Littlewood–Richardson coefficients are log-concave sequences with no internal zeros.","Mixed products of ordinary skew Schur, skew P, and skew Q functions again have realizable factorial normalizations, so every inequality and support statement extends to products."],"supporting_citations":[{"why":"Posits the two Lorentzian conjectures that the paper settles in stronger volume form.","marker":"[15]"},{"why":"Supplies the total-Chern-to-Lorentzian mechanism used for the direct Lorentzian implications.","marker":"[9]"},{"why":"Supplies the realizable-covolume operator theorem that removes the factorial shifts and upgrades to realizable volume polynomials.","marker":"[13]"},{"why":"Gives the modified Pfaffian representatives that match Schur Q-functions to Lagrangian Schubert classes, the normalization-sensitive step.","marker":"[19]"},{"why":"Supplies the stable theta-polynomial Schubert homomorphism whose k=0 case is the type-C ring map.","marker":"[5]"},{"why":"Provides the Schur P/Q symmetric-function conventions and the factorization identity underlying the type-C coefficient formula.","marker":"[23]"},{"why":"Supplies the projective-bundle and intersection-theoretic formulas used to build explicit volume realizations.","marker":"[10]"},{"why":"Provides the Richardson variety dimension and intersection class used in both type A and type C skew identities.","marker":"[6]"},{"why":"Defines Lorentzian polynomials and proves that top intersections of nef divisors are Lorentzian.","marker":"[4]"}],"fun_headline_variants":["All skew Schur and Schur P/Q functions are volume polynomials","Volume realizations settle skew Schur and Schur P/Q conjectures","Every skew Schur and Schur P/Q function is a volume polynomial","Skew Schur and Schur P/Q functions are realizable volume polynomials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The shifted theorem depends on a matching between algebraic representatives of Schur Q-functions and geometric Schubert classes on the Lagrangian Grassmannian; if that matching is not exactly right, the skew P/Q volume conclusions would not follow.","fun_headline_variants_meta":{"raw":{"variants":["All skew Schur and Schur P/Q functions are volume polynomials","Volume realizations settle skew Schur and Schur P/Q conjectures","Every skew Schur and Schur P/Q function is a volume polynomial","Skew Schur and Schur P/Q functions are realizable volume polynomials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000738,"raw_usage":{"total_tokens":3383,"prompt_tokens":1115,"completion_tokens":2268,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":731,"completion_tokens_details":{"reasoning_tokens":2189}},"tokens_in":731,"tokens_out":2268,"duration_ms":14063,"temperature":1.0,"reasoning_tokens":2189,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:20:13.287385+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small strict skew shape, for instance $\\lambda=(2,1)$, $\\mu=\\emptyset$, with $n=2$ and $\\alpha=(2,1)$. Compute $[x^\\alpha]P_{\\lambda/\\mu}$ by the marked shifted-tableau generating function and separately compute $\\int_{R^C_{\\lambda/\\mu}} c_2(E)c_1(E)$ on the Lagrangian Richardson variety; the paper's theorem says the two numbers are equal (both equal 1). Any mismatch in such a finite computation would falsify the type-C volume theorem, and the same check can be run for ordinary skew Schur polynomials with the type-A Richardson identity.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Posits the two Lorentzian conjectures that the paper settles in stronger volume form."},{"cited_title":"Mixed Segre zeta functions and their log-concavity","cited_arxiv_id":"2507.06424","evidence_quote":"Supplies the total-Chern-to-Lorentzian mechanism used for the direct Lorentzian implications."},{"cited_title":"Kresch and H","cited_arxiv_id":null,"evidence_quote":"Gives the modified Pfaffian representatives that match Schur Q-functions to Lagrangian Schubert classes, the normalization-sensitive step."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the stable theta-polynomial Schubert homomorphism whose k=0 case is the type-C ring map."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Schur P/Q symmetric-function conventions and the factorization identity underlying the type-C coefficient formula."},{"cited_title":"Fulton,Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, vol","cited_arxiv_id":null,"evidence_quote":"Supplies the projective-bundle and intersection-theoretic formulas used to build explicit volume realizations."},{"cited_title":"Brion,Lectures on the geometry of flag varieties, in Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, 2005, pp","cited_arxiv_id":null,"evidence_quote":"Provides the Richardson variety dimension and intersection class used in both type A and type C skew identities."},{"cited_title":"Brändén and J","cited_arxiv_id":null,"evidence_quote":"Defines Lorentzian polynomials and proves that top intersections of nef divisors are Lorentzian."}],"review_version":1}