Postnikov--Stanley polynomials are Lorentzian

Katherine Tung; Serena An; Yuchong Zhang

Postnikov--Stanley polynomials are Lorentzian

T0 review · 0 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Every Postnikov–Stanley polynomial attached to a Bruhat interval in any Weyl group is Lorentzian.

desk verdict Short, correct proof that Postnikov–Stanley polynomials are Lorentzian via identification with Richardson variety degree polynomials; the new geometric step holds up. read the letter →

arxiv 2412.02051 v2 pith:M5JIRQAO submitted 2024-12-03 math.CO math.AG

Serena An , Katherine Tung , Yuchong Zhang This is my paper · ORCID

classification math.COmath.AG MSC 05E1414M1505A20

keywords Postnikov-StanleypolynomialsLorentzianRichardsonvarietiesSchubertBruhatorderM-convexsupportNewton-Okounkovbodiesvolume

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

Soft 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.

Who 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.

Referee Report

0 major / 3 minor

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.

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.

minor comments (3)

[Section 2.4, Proposition 2.5] 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 4, proof of Theorem 1.2] 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 3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Postnikov–Stanley identity is proved from the Chevalley formula and the Lorentzian step from the volume-polynomial theorem; self-citation is limited to the corollary being resolved.

full rationale

The derivation chain is self-contained relative to standard external geometric inputs. The central identity (Proposition 3.3) is proved by expanding λ^ℓ · σ_u via the Chevalley formula (Lemma 3.5, cited from Chevalley) and pairing with the Richardson class formula from Speyer [43]; no parameter is fitted to D_u^w and no expression is defined in terms of the target polynomial. The Lorentzian step applies the Brändén–Huh volume polynomial theorem (Theorem 2.8) to nef divisors on the Richardson variety R_u^w, with nefness imported from standard facts (Hague [20] for flag varieties and Lazarsfeld Example 1.4.4 for restrictions). The scalar 1/(ℓ(w)−ℓ(u))! is positive, so multiplying the volume polynomial preserves Lorentzianity. The self-citation [2] (An–Tung–Zhang 2024) appears only as the conjecture being resolved in Corollary 1.3, not as an assumption used in the proof. No equation reduces to its own input, and no fitted quantity is renamed as a prediction. The quoted geometric inputs, including Proposition 2.5, are external results for finite Weyl groups rather than self-citations, so they do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof rests entirely on standard Schubert calculus and the Brändén-Huh volume polynomial theorem. There are no fitted parameters and no new postulated objects. The main input from the authors' own prior work is the conjecture that Corollary 1.3 resolves, not a lemma used to prove it.

assumptions (5)

standard math Chevalley multiplicity formula λ·σ_w = Σ_{α: ℓ(ws_α)=ℓ(w)+1} (λ,α∨) σ_{ws_α} (Lemma 3.5, cited from Chevalley).
This is the bridge between Bruhat-order chain weights and multiplication by divisor classes in H*(G/B); the paper cites it without proof.
domain assumption Richardson class formula [R_u^w] = σ_{w0w}·σ_u and irreducibility of R_u^w (Proposition 2.5, cited from Speyer).
Needed to express the λ-degree of R_u^w as a pairing of Schubert classes; if the product formula failed, Proposition 3.3 would fail.
domain assumption Nefness criterion: on G/B, L_λ is nef if and only if λ is dominant (Lemma 4.1, cited from Hague).
Used to ensure the fundamental divisors restrict to nef divisors on R_u^w, the hypothesis of the volume polynomial theorem.
standard math Theorem 2.8: the volume polynomial of nef Cartier divisors on an irreducible projective variety is Lorentzian (Brändén-Huh).
This external theorem is the engine that converts the degree-polynomial identity into the Lorentzian conclusion; the paper does not reprove it.
standard math Poincaré pairing orthonormality ⟨σ_u, σ_{w0w}⟩ = δ_{u,w} (Lemma 3.7, cited from Postnikov-Stanley).
Used to pick out the D_u^w term from the sum over w' in Proposition 3.3.

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Pith. "Pith review of Postnikov--Stanley polynomials are Lorentzian." pith.science (2026). https://pith.science/paper/M5JIRQAO

@misc{pith2026241202051,
  author       = {Pith},
  title        = {Pith review of: Postnikov--Stanley polynomials are Lorentzian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5JIRQAO}},
  note         = {Machine review of arXiv:2412.02051}
}

abstract

Postnikov--Stanley polynomials $D_u^w$ are a generalization of skew dual Schubert polynomials to the setting of arbitrary Weyl groups. We prove that Postnikov--Stanley polynomials are Lorentzian by showing that they are degree polynomials of Richardson varieties. Our result yields an interesting class of Lorentzian polynomials related to the geometry of Richardson varieties, generalizes the result that dual Schubert polynomials are Lorentzian (Huh--Matherne--M\'esz\'aros--St. Dizier 2022), and resolves the conjecture that Postnikov--Stanley polynomials have M-convex support (An--Tung--Zhang 2024).

Figures

Figures reproduced from arXiv: 2412.02051 by the authors.

**Figure 1.** Calculating the Postnikov–Stanley polynomial D321 213 in A2. For u ≤ w in the Bruhat order of W, the (Bruhat) interval [u, w] is the subposet containing all v ∈ W such that u ≤ v ≤ w. 2.3. Postnikov–Stanley polynomials. We defined Postnikov–Stanley polynomials Dw u for u ≤ w ∈ W in Definition 1.1. The skew dual Schubert polynomials are Postnikov–Stanley polynomials for which W is of type A, and the dual Schubert pol… view at source ↗

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Supersymmetric Schur polynomials have saturated Newton polytopes
math.CO 2025-07 reject novelty 5.0 of 10

A proof that supersymmetric Schur polynomials have SNP is invalid because the stated hook-inequality support set is contradicted by the paper's own example and by symmetry.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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Toric degenerations of spherica l varieties

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