{"id":"34b5f9ed-890e-4f1f-995a-ba3d7f5232d6","arxiv_id":"2412.02051","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Postnikov-Stanley polynomials are Lorentzian because they are, up to a factorial factor, the degree polynomials of Richardson varieties.","lead":"The authors prove that Postnikov-Stanley polynomials, a generalization of skew dual Schubert polynomials to arbitrary Weyl groups, are Lorentzian. The proof identifies these polynomials, up to a factorial factor, with the degree polynomials of Richardson varieties, which also resolves a conjecture on M-convex support.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the Richardson-class and nefness inputs are standard, and the volume-polynomial argument is sound.","rationale":"The reader's weakest assumption correctly identifies the most cited geometric input: the class formula [R_u^w] = [X^w] * [X_u] and the irreducibility/dimension of Richardson varieties. I agree that these are the least re-derived parts of the argument. However, on inspection they are established facts in the Schubert calculus of finite Weyl groups, and the paper's use of them is faithful. The volume-polynomial theorem of Branden and Huh is stated for irreducible projective varieties without a smoothness hypothesis, and the restriction of nef line bundles to a subvariety is nef, so the passage from G/B to the possibly singular Richardson variety is justified. The proof of Proposition 3.3 is a direct computation with Chevalley's formula and the Poincare pairing, and the scaling by a positive factorial preserves the Lorentzian property. I found no internal inconsistency, no circular step, and no unsupported assumption that changes the soundness of the main theorem. A fully self-contained paper might prove Proposition 2.5 or at least state the precise transversality result, but the citation is to a standard reference and does not introduce correctness risk at the level of the central claim.","tokens_in":9748,"tokens_out":15448,"duration_ms":163914,"concrete_test":"Run an independent check of Proposition 3.3 in a non-type-A case, e.g. W = B2, u = s1, w = s2s1: compute ell! * D_u^w(lambda) from saturated chains using Chevalley multiplicities, and compute the degree of R_u^w via the known product sigma_{w0w} * sigma_u in Schubert cohomology. If the two disagree for a dominant weight lambda, the geometric identification, and hence Theorem 1.2, fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the proof as a whole and found no load-bearing flaw. The central identification (Proposition 3.3) depends on the Richardson class formula [R_u^w] = sigma_{w0w} * sigma_u and on R_u^w being an irreducible projective variety of dimension ell(w)-ell(u); both are standard results for finite Weyl groups, cited from Speyer [43] and the Richardson variety literature. The subsequent Lorentzian step is also sound: restricting the nef line bundles L_{omega_i} to R_u^w preserves nefness (Section 4, citing Lazarsfeld Example 1.4.4), and the Branden-Huh volume-polynomial theorem applies to irreducible projective varieties, so no smoothness or normality of R_u^w is needed. The scalar 1/(ell(w)-ell(u))! is positive, hence preserves Lorentzianity. The only minor gap is that Proposition 2.5 is quoted rather than proved; this is a citation-level omission, not a correctness risk. I therefore treat the reader's weakest assumption as a nontrivial but standard geometric input, not a reason to lower the verdict.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that Postnikov--Stanley polynomials $D_u^w$, defined for arbitrary Weyl groups as normalized sums over saturated Bruhat chains, are Lorentzian. The proof has two steps. First, via the Chevalley formula (Lemma 3.5) and Poincar\\'e duality, the authors identify $D_u^w$ with the degree polynomial of the Richardson variety $R_u^w = X^w \\cap X_u$ up to the factor $(\\ell(w)-\\ell(u))!$ (Proposition 3.3). Second, using the fact that the restrictions of the line bundles $L_{\\omega_i}$ to $R_u^w$ are nef, they invoke the Br\\\"and\\'en--Huh volume-polynomial theorem (Theorem 2.8) to conclude that the degree polynomial, hence $D_u^w$, is Lorentzian. The paper also derives the M-convex support of Postnikov--Stanley polynomials as an immediate corollary, resolving a conjecture of An--Tung--Zhang.","tokens_in":9901,"tokens_out":18013,"duration_ms":185192,"significance":"The result is significant: it provides a large new family of Lorentzian polynomials tied to Richardson varieties, generalizes the previously known Lorentzianity of dual Schubert polynomials, and resolves the M-convex support conjecture. The proof is short and transparent, and the key identification with volume polynomials is a clean geometric explanation rather than an ad hoc analytic verification. The authors correctly isolate the standard geometric inputs: the Richardson class formula, nefness of fundamental line bundles, and the Br\\\"and\\'en--Huh theorem. No circularity or post hoc fitting is present. The paper is a strong contribution to the interface of algebraic combinatorics and algebraic geometry.","major_comments":[],"minor_comments":[{"comment":"The parenthetical 'Since the intersection $X^w \\cap X_u$ is transverse' is not accurate as stated: Richardson varieties can be singular, and the Schubert and opposite Schubert varieties need not meet transversely at every point. What is needed is that the intersection is proper, reduced, and irreducible, with the stated cohomology class. Please rephrase this sentence and explicitly record that $R_u^w$ is an irreducible projective variety of dimension $\\ell(w)-\\ell(u)$, citing a standard reference for this fact in addition to the class formula cited from [43].","section":"Section 2.4, Proposition 2.5"},{"comment":"The notation '$D'_i := D_i \\cap R_u^w$ be the Cartier divisor corresponding to $L_{\\omega_i}|_{R_u^w}$' is not literally correct when $R_u^w$ is contained in the support of $D_i$, as can happen for Schubert divisors. Since every line bundle on an irreducible projective variety is the line bundle of some Cartier divisor, one should instead fix Cartier divisors $D'_i$ on $R_u^w$ with $O(D'_i) \\cong L_{\\omega_i}|_{R_u^w}$; their nefness follows from the same cited result. This is a local expository fix and does not affect the validity of the argument.","section":"Section 4, proof of Theorem 1.2"},{"comment":"The symbol $\\ell$ is used both for the Coxeter length function and for the integer exponent in Lemma 3.6 and Proposition 3.3. Using a separate symbol such as $d$ for the exponent would improve readability and avoid possible confusion.","section":"Section 3"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The punchline: this is a short, correct paper that proves what it says, and the genuinely new part is the identification of Postnikov–Stanley polynomials with degree polynomials of Richardson varieties. Everything downstream—Lorentzianity from Brändén–Huh—is then routine, but the identification is real and covers all Weyl groups and all u ≤ w, not just the type-A case from Postnikov–Stanley.\n\nWhat it does well: the proof is transparent. Lemma 3.6 is a clean induction from the Chevalley formula, Proposition 3.3 follows by Poincaré duality, and the volume-polynomial argument in Section 4 is exactly the right tool. The paper also gives proper credit: it cites [39, Prop 4.2] as the direct ancestor and [23, Prop 18] as the type-A dual Schubert case, and it resolves the authors’ own M-convex support conjecture without using that conjecture as input. No fitting, no circularity.\n\nSoft spots, in proportion: they are minor. The geometric inputs—Richardson class formula [R_u^w] = σ_{w0w}·σ_u, nefness of restrictions, and applicability of the volume polynomial theorem to singular Richardson varieties—are cited rather than proved. That is normal for this literature, and the stress-test concern about transversality/irreducibility is real but standard: the intersection defining R_u^w is transverse and the variety is irreducible, both known results. The only genuine exposition gap I see is in Proposition 3.3: the step from ⟨σ_{w0w}·σ_u, λ^ℓ⟩ to the coefficient of σ_{w0} is fine but could use one sentence saying the pairing picks out the top class. A referee should ask for that, nothing more.\n\nWho it is for: algebraic combinatorialists working on Lorentzian polynomials or Schubert calculus, and anyone tracking the Newton polytope/M-convexity program. It will be cited. I would send it to peer review; the theorem is worth a serious referee, and the paper is ready after small edits.","headline":"Short, correct proof that Postnikov–Stanley polynomials are Lorentzian via identification with Richardson variety degree polynomials; the new geometric step holds up.","tokens_in":10481,"tokens_out":1565,"would_cite":true,"duration_ms":15564,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E14","14M15","05A20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every Postnikov–Stanley polynomial attached to a Bruhat interval in any Weyl group is Lorentzian.","keywords":["Postnikov-Stanley polynomials","Lorentzian polynomials","Richardson varieties","Schubert polynomials","Bruhat order","M-convex support","Newton-Okounkov bodies","volume polynomials"],"falsifier":"The theorem would be refuted by exhibiting any pair $u\\le w$ in any Weyl group for which the support of $D_u^w$ is not M-convex, or for which some second derivative of $D_u^w$ has more than one positive eigenvalue. Since Lorentzianness is decidable by finite computation from the Bruhat interval, a search through rank-two and rank-three Weyl groups would be enough to look for such a counterexample.","tokens_in":9507,"feed_emoji":"📐","tokens_out":10421,"duration_ms":98225,"temperature":0.7,"pith_summary":"Postnikov–Stanley polynomials $D_u^w$ are homogeneous polynomials associated to every interval $[u,w]$ in the Bruhat order of a Weyl group; they generalize skew dual Schubert polynomials. This paper proves that every one of them is Lorentzian: nonnegative coefficients, M-convex support, and log-concavity under every sequence of coordinate derivatives. The proof shows the polynomial is the degree polynomial of the Richardson variety $R_u^w$, so that up to the factorial factor $(\\ell(w)-\\ell(u))!$, its value at a dominant weight counts intersection points of the variety with a generic linear subspace. Because these degrees are volume polynomials of nef divisors, the Lorentzian property follows from the standard volume-polynomial theorem. The M-convex support conjecture for Postnikov–Stanley polynomials is then a corollary.","feed_headline":"Proved: every Postnikov–Stanley polynomial is Lorentzian","feed_subtitle":"A geometric degree identity ties them to Richardson varieties, yielding log-concavity and M-convex support.","key_machinery":"The load-bearing object is the Richardson variety $R_u^w=X^w\\cap X_u$ and its degree polynomial. The central identity is $$\\deg_\\$\\lambda$(R_u^w)=(\\ell(w)-\\ell(u))!\\,D_u^w(\\$\\lambda$),$$ where the degree is the number of points in the intersection of the embedded variety with a generic linear subspace of complementary codimension. The identity is assembled from three ingredients: the Chevalley formula multiplying a Schubert class by a hyperplane class, the Richardson class formula $[R_u^w]=\\sigma_{w_0w}\\cdot\\sigma_u$, and the Poincaré pairing on Schubert classes. Once $D_u^w$ is recognized as a degree polynomial, the theorem that volume polynomials of nef divisors are Lorentzian applies verbatim, because the restrictions of the line bundles $L_{\\omega_i}$ to $R_u^w$ are nef.","core_discovery":"The core claim, on the paper's own terms, is that the Postnikov–Stanley polynomial $D_u^w$ equals, up to the constant $(\\ell(w)-\\ell(u))!$, the $\\lambda$-degree of the Richardson variety $R_u^w=X^w\\cap X_u$ in the flag variety of the corresponding simply connected group. The paper proves this geometric identity by expanding $\\lambda^\\ell\\cdot\\sigma_u$ along saturated chains via the Chevalley formula and pairing against the Richardson class $\\sigma_{w_0w}\\cdot\\sigma_u$. Applying the theorem that volume polynomials of nef divisors are Lorentzian then yields the theorem for arbitrary Weyl groups, and the corollary settles the conjecture that these polynomials have M-convex support.","pith_inferences":["If the same degree-polynomial argument works for projected Richardson varieties or for Richardson varieties in other flag varieties, it would produce new Lorentzian families; the paper does not claim this.","Because Lorentzian polynomials are closed under multiplication, one could ask whether products of Postnikov–Stanley polynomials from compatible intervals remain Lorentzian; this is a testable extension rather than a result of the paper.","The Newton–Okounkov body identification suggests that explicit polytopal models of these bodies could yield purely combinatorial proofs of M-convexity; the paper does not construct such models.","The theorem may have algorithmic use: checking that supports are M-convex is finite for fixed rank, so the conjecture could be verified computationally in new Lie types before a geometric proof is known; this is extrapolation from the paper's method."],"forward_implications":["Dual Schubert polynomials are Lorentzian: taking $W$ of type A and $u=\\mathrm{id}$ recovers the earlier result for dual Schubert polynomials.","Every $D_u^w$ has M-convex support, so its Newton polytope is a saturated generalized permutahedron; this resolves the M-convex support conjecture.","Every positive derivative of $D_u^w$ is log-concave on the positive orthant, and any top-degree derivative has at most one positive eigenvalue; this is exactly the content of being Lorentzian.","The volume of the Newton–Okounkov body $\\Delta_{w_0}(R_u^w,\\lambda)$ equals $D_u^w(\\lambda)$, giving a convex-geometric interpretation of the Bruhat-chain sums.","Evaluating at a dominant weight $\\lambda$ counts, up to the factorial factor, the number of points of $R_u^w$ in a generic linear subspace of complementary dimension."],"supporting_citations":[{"why":"Supplies the theorem that volume polynomials of nef divisors are Lorentzian, the main engine of the proof.","marker":"[8]"},{"why":"Defines the Postnikov–Stanley polynomials and provides the Schubert-variety degree formula and pairing lemma that Proposition 3.3 generalizes.","marker":"[39]"},{"why":"Provides the Richardson class product formula $[R_u^w]=\\sigma_{w_0w}\\cdot\\sigma_u$, the geometric input that identifies $D_u^w$ with a degree.","marker":"[43]"},{"why":"Supplies the Chevalley formula used to expand a hyperplane class times a Schubert class along Bruhat covers.","marker":"[14]"},{"why":"Shows that the line bundle $L_\\lambda$ is nef exactly for dominant $\\lambda$, making the restricted divisors nef.","marker":"[20]"},{"why":"Establishes the type-A, $u=\\mathrm{id}$ special case that the main theorem generalizes.","marker":"[23]"},{"why":"States the M-convex support conjecture that the corollary resolves.","marker":"[2]"},{"why":"Identifies the degree with the volume of the Newton–Okounkov body in Remark 3.8, giving the convex-geometric reading of $D_u^w$.","marker":"[24]"}],"fun_headline_variants":["Postnikov–Stanley polynomials are Lorentzian via Richardson varieties","Lorentzian proof: Postnikov–Stanley polynomials from Richardson degrees","Arbitrary Weyl groups: Postnikov–Stanley polynomials are Lorentzian","M-convex support and Lorentzian: Postnikov–Stanley resolved","Geometric degree identity shows Postnikov–Stanley Lorentzian"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the quoted geometric fact that each Richardson variety $R_u^w$ is irreducible and has cohomology class $[R_u^w]=\\sigma_{w_0w}\\cdot\\sigma_u$; if the intersection failed to be transverse or irreducible for some $u,w$, the degree-polynomial identity and the Lorentzian conclusion would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Postnikov–Stanley polynomials are Lorentzian via Richardson varieties","Lorentzian proof: Postnikov–Stanley polynomials from Richardson degrees","Arbitrary Weyl groups: Postnikov–Stanley polynomials are Lorentzian","M-convex support and Lorentzian: Postnikov–Stanley resolved","Geometric degree identity shows Postnikov–Stanley Lorentzian"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000661,"raw_usage":{"total_tokens":2948,"prompt_tokens":797,"completion_tokens":2151,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":413,"completion_tokens_details":{"reasoning_tokens":2056}},"tokens_in":413,"tokens_out":2151,"duration_ms":18581,"temperature":1.0,"reasoning_tokens":2056,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:54:12.346560+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The theorem would be refuted by exhibiting any pair $u\\le w$ in any Weyl group for which the support of $D_u^w$ is not M-convex, or for which some second derivative of $D_u^w$ has more than one positive eigenvalue. Since Lorentzianness is decidable by finite computation from the Bruhat interval, a search through rank-two and rank-three Weyl groups would be enough to look for such a counterexample.","supporting_citations":[{"cited_title":"Lorentzian polynomials.Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that volume polynomials of nef divisors are Lorentzian, the main engine of the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Postnikov–Stanley polynomials and provides the Schubert-variety degree formula and pairing lemma that Proposition 3.3 generalizes."},{"cited_title":"Richardson varieties, projected Richardson v arieties, and positroid varieties, 2023","cited_arxiv_id":null,"evidence_quote":"Provides the Richardson class product formula $[R_u^w]=\\sigma_{w_0w}\\cdot\\sigma_u$, the geometric input that identifies $D_u^w$ with a degree."},{"cited_title":"Chevalley","cited_arxiv_id":null,"evidence_quote":"Supplies the Chevalley formula used to expand a hyperplane class times a Schubert class along Bruhat covers."},{"cited_title":"Cohomology of flag varieties and the BK-filtration","cited_arxiv_id":null,"evidence_quote":"Shows that the line bundle $L_\\lambda$ is nef exactly for dominant $\\lambda$, making the restricted divisors nef."},{"cited_title":"Matherne, Karola M´ esz´ aros, and Avery St","cited_arxiv_id":null,"evidence_quote":"Establishes the type-A, $u=\\mathrm{id}$ special case that the main theorem generalizes."},{"cited_title":"Newton polyto pes of dual Schubert polynomials, 2024","cited_arxiv_id":null,"evidence_quote":"States the M-convex support conjecture that the corollary resolves."},{"cited_title":"Note on cohomology rings of spherical varieties and volume polynomial","cited_arxiv_id":null,"evidence_quote":"Identifies the degree with the volume of the Newton–Okounkov body in Remark 3.8, giving the convex-geometric reading of $D_u^w$."}],"review_version":1}