Newton polytopes of dual Schubert polynomials

Katherine Tung; Serena An; Yuchong Zhang

Newton polytopes of dual Schubert polynomials

T0 review · 0 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Inversions determine every dual Schubert polynomial's support.

desk verdict New support and vertex characterizations for dual Schubert polynomials, with a clean elementary proof that is mostly solid; Lemma 3.7 needs a fuller proof but the gap is fillable. read the letter →

arxiv 2411.16654 v1 pith:S72AHT2E submitted 2024-11-25 math.CO

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

classification math.CO MSC 05E0505E1452B1205A05

keywords dualSchubertpolynomialsNewtonpolytopesM-convexitysaturatedpolytopeBruhatordergreedychainsgeneralizedpermutahedrastaircaseYoungdiagrams

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

The paper proves that the support of a dual Schubert polynomial $D_w$—the set of exponent vectors appearing in its monomials—is completely determined by the inversion pairs of the permutation $w$. Concretely, $\operatorname{supp}(D_w)$ is the Minkowski sum, over all inversions $(a,b)$ of $w$, of the intervals $\{e_a,\ldots,e_{b-1}\}$, which is exactly the support of the product $\operatorname{GW}(w)$ of linear forms $x_a+\cdots+x_{b-1}$. This yields an elementary proof that $D_w$ is M-convex and that its Newton polytope is a generalized permutahedron, and it identifies the vertices of that polytope as the monomials with coefficient $1$ in $\operatorname{GW}(w)$. Because dual Schubert polynomials encode degrees of Schubert varieties and pair naturally with Schubert polynomials, knowing their Newton polytopes exactly makes those invariants directly readable from the inversion set.

What carries the argument

The load-bearing object is the greedy chain: a saturated chain $\mathrm{id}=w_0\lessdot w_1\lessdot\cdots\lessdot w_\ell=w$ in the strong Bruhat order such that each transposition $w_i=w_{i-1}t_{ab}$ cannot be replaced by a transposition with a wider interval $[a',b']$ while staying inside $[\mathrm{id},w]$. The paper defines the global weight $\operatorname{GW}(w)$ as the product over inversion pairs $(a,b)$ of $x_a+x_{a+1}+\cdots+x_{b-1}$. It proves that every greedy chain has weight $\operatorname{GW}(w)$ and that every Bruhat interval contains at least one greedy chain; together with the inclusion that every chain's support is contained in $\operatorname{supp}(\operatorname{GW}(w))$, this single-chain comparison equates $\operatorname{supp}(D_w)$ with $\operatorname{supp}(\operatorname{GW}(w))$.

What would settle it

Compute $D_w$ and $\operatorname{GW}(w)$ explicitly for all $w$ in $S_6$ and compare supports; any monomial in one support but not the other would refute Theorem 1.2. Alternatively, exhibit a Bruhat interval with no greedy chain, directly contradicting Lemma 3.7.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the averaging over saturated chains that defines $D_w$ can be replaced by a single product: for the greedy chains introduced here, the weight of every greedy chain in $[\mathrm{id}, w]$ is exactly $\operatorname{GW}(w)$, and every Bruhat interval contains at least one greedy chain. Since every chain's weight has support contained in $\operatorname{supp}(\operatorname{GW}(w))$, the support of the averaged polynomial $D_w$ coincides with $\operatorname{supp}(\operatorname{GW}(w))$, giving Theorem 1.2. A further lemma then shows that the vertices of $\operatorname{Newton}(D_w)$ are precisely the monomials of $\operatorname{GW}(w)$ with coefficient $1$, so the polytope's extremal structure is a purely combinatorial artifact of the inversion set.

Load-bearing premise

For the equality of supports, the proof needs the claim that every Bruhat interval contains a chain whose transpositions are all as wide as possible; this is asserted with only a one-sentence construction, and if some interval lacks such a chain the identity could fail.

Editorial extensions

If this is right

$D_w$ is M-convex, so its Newton polytope is a generalized permutahedron; the paper gives explicit inequality data $z_I=\sum_{(a,b)\in\operatorname{Inv}(w)} \mathbf{1}_{I\supseteq[a,b)}$ for that polytope.
The vertices of $\operatorname{Newton}(D_w)$ can be listed combinatorially by rectangle tilings of a staircase Young diagram labelled by inversion pairs, giving a purely diagrammatic enumeration.
Every monomial of $D_w$ appears in $\operatorname{GW}(w)$, so support comparisons between different permutations can be studied through inversion-set inclusion rather than chain enumeration.
The proof of M-convexity is elementary and does not rely on the algebraic-geometric machinery of the earlier proof.

Reading between the lines

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

If the greedy-chain existence lemma could be strengthened to intervals $[u,w]$ with an arbitrary lower bound, the same argument would give support formulas for Postnikov–Stanley polynomials, which the paper leaves as a conjecture about M-convexity.
The vertex characterization suggests that linear optimization over $\operatorname{Newton}(D_w)$ is governed by a greedy rectangle-tiling rule, which may yield a fast algorithm for evaluating Schubert-degree-type quantities.
The paper's matroidal decomposition of $\operatorname{Newton}(D_w)$ as a Minkowski sum of rank-one matroid polytopes might transfer to other families of interval-defined polynomials, where a similar support identity could be tested.

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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

Best read of this: the paper actually delivers what it advertises. The new content is Theorem 1.2, the support identity supp(D_w) = Minkowski sum of inversion intervals, equivalently supp(GW(w)), and Corollary 1.4 identifying vertices of Newton(D_w) as the coefficient-one monomials of GW(w). M-convexity itself was known from Huh–Matherne–Mészáros–St. Dizier, but their proof is Lorentzian/algebraic; this one is a direct chain argument and it gives more precision, not just a rerun.

The main line is sound. Lemma 3.13 and Lemma 3.9 are the load-bearing parts; I checked the inductions and the inversion-set analysis. The inclusion supp(D_w) ⊆ supp(GW(w)) follows from the dominant pairing in Lemma 3.13 and is clean. The reverse inclusion depends on Lemma 3.7, existence of a greedy chain in every Bruhat interval. That lemma has to carry more weight than its proof does. The text gives a one-sentence top-down construction and doesn't spell out why the chosen cover exists with the required maximality, nor why all earlier edges remain greedy. As the stress-test note says, the existence claim is true — choose a cover whose swapped pair is inclusion-maximal among covers in the interval at each step — so this is a presentation gap, not a correctness flaw. But for a referee, I'd want Lemma 3.7 expanded before publication.

The generalized permutahedron formulation in Theorem 3.18 and the rectangle-tiling vertex description are useful additions. The examples check out, and the explicit vertex computation for D_{253641} is consistent with the coefficient-one characterization. No circularity: the proof never assumes M-convexity or SNP; it derives them from the chain-weight argument. The conjectures in Section 5 are marked as conjectures, appropriately.

Who this is for: algebraic combinatorialists working on Schubert polynomials, Newton polytopes, and M-convexity. It is a strengthening of a known result rather than a breakthrough, but it gives a genuinely simpler proof and a sharper statement. I would send this to a serious referee; the missing details in Lemma 3.7 are fixable and everything else holds up.

Referee Report

0 major / 5 minor

Summary. The paper studies dual Schubert polynomials D_w, defined as averages of weights of saturated chains in the Bruhat interval [id, w]. Its main result (Theorem 1.2) identifies supp(D_w) with the Minkowski sum of elementary-vector sets indexed by the inversions of w, equivalently with supp(GW(w)), where GW(w) is the product over inversion pairs (a,b) of the linear forms x_a + ... + x_{b-1}. From this support identity the authors derive an elementary proof of M-convexity (Corollary 1.3), describe Newton(D_w) as a generalized permutahedron with explicit parameters (Theorem 3.18), and characterize its vertices as the monomials of coefficient 1 in GW(w) (Corollary 1.4), together with a staircase-tiling description.

Significance. If correct, this is a genuine strengthening of the Huh--Matherne--Mészáros--St. Dizier M-convexity theorem: the support is made explicit, the Newton polytope is identified as a generalized permutahedron with a simple parameter formula, and the vertices are characterized combinatorially. A particular strength is that the proof is self-contained: it does not invoke the earlier M-convexity result, and it proceeds through an elementary greedy-chain argument and the global weight GW(w). The matroid-polytope passage in Section 3.3 is clean and uses only standard facts. The conjectures about Postnikov--Stanley polynomials are clearly labeled as such, and the computational checks in SageMath are an honest part of the evidence. Overall, the paper is a solid contribution to the Newton-polytope literature in algebraic combinatorics.

minor comments (5)

[Lemma 3.7] The proof of Lemma 3.7 is a single sentence and is the only place where the reverse inclusion supp(GW(w)) ⊆ supp(D_w) is justified. The construction of a maximal cover at each top-down step should be written out, and the proof should state explicitly why the greedy condition for previously constructed edges is preserved when the interval [u,w] is fixed. The assertion is true, but as written it is too terse for a load-bearing lemma.
[Example 4.2 / Figure 2] The permutation in Example 4.2 is written as D254361 in the text and as D253641 in Step 2 and the figure caption; the two notations should be reconciled.
[Theorem 3.18] D_w is a polynomial in n-1 variables, but the matroid polytopes P(M_ab) in Equation (1) live in R^n. The proof should state explicitly that supports and Newton polytopes are embedded in R^n with the last coordinate equal to zero, so that the Minkowski-sum calculation is formally consistent.
[Section 4, Step 4] The procedure states that each rectangle sum is written at the bottom right corner of the rectangle, but in Figures 2 and 3 the sums appear in the bottom rows and are then read top to bottom; a sentence clarifying the reading order and how the displayed vertex coordinates are formed would improve reproducibility.
[Remark after Example 3.2] The remark asserts, based on SageMath, that D^{4231}_{1324} has SNP but not SCNP; since this is used to explain why the SCNP method does not generalize, it would be helpful to include the exact polynomial or the computation script in an ancillary file.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the support identity is derived from Bruhat-order lemmas and external polytope facts, not assumed.

full rationale

The central claim supp(D_w) = supp(GW(w)) is not built into any definition. D_w is defined as an average over all saturated chains, while GW(w) is defined independently as the product of the inversion linear forms. The forward inclusion uses Lemma 3.13 (generating set domination) and Lemma 3.14, together with Proposition 2.7 on supports of products; none of these assumes the target equality. The reverse inclusion uses Lemma 3.9, which proves by induction that the weight of any greedy chain equals GW(w), and Lemma 3.7, which asserts existence of a greedy chain. Lemma 3.7's proof is one sentence and terse, but it is an independent combinatorial existence claim; if it were false the theorem would fail, which would be a correctness issue, not circularity. No parameter is fitted to the support data, and no subset of the claimed support identity is used as an input. The M-convexity conclusion is derived after the support identity using standard external facts about matroid polytopes, generalized permutahedra, and Murota's characterization, not taken from Huh et al. The Huh et al. result is cited only as context and as a comparison point. The vertex characterization invokes external results [1, Theorem 3.5] and [14, Corollary 8.2], neither of which is supplied by the present authors. There are no load-bearing self-citations, no fitted inputs renamed as predictions, and no uniqueness theorem imported from the authors' prior work. The derivation chain is self-contained modulo standard Bruhat-order and polytope facts.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no free parameters and no new postulated entities. It relies on standard cited results: the Postnikov-Stanley chain formula, support/Newton behavior under multiplication, matroid polytope facts, M-convex equivalence, and vertex theorems of Agnarsson-Morris and Postnikov. All are published external results.

assumptions (8)

standard math Postnikov-Stanley formula: D_u^w is the weighted sum over saturated chains from u to w divided by (ℓ(w)-ℓ(u))! (Definition 2.2).
Definition of the object studied; from Postnikov and Stanley [15].
standard math For f,g with nonnegative coefficients, supp(fg)=supp(f)+supp(g) and supp(f+g)=supp(f)∪supp(g) (Proposition 2.7).
Elementary fact used throughout to convert chain supports into products.
standard math Newton polytopes respect products and sums: Newton(fg)=Newton(f)+Newton(g), Newton(f+g)=conv(Newton(f)∪Newton(g)) (Proposition 2.8, Monical-Tokcan-Yong).
Used to identify Newton(D_w) with the Minkowski sum of matroid polytopes.
standard math A homogeneous polynomial is M-convex iff it has SNP and its Newton polytope is a generalized permutohedron (Proposition 2.18, Murota).
Bridges the support property to M-convexity in Corollary 1.3.
standard math Matroid polytopes are generalized permutohedra with halfspace descriptions, and Minkowski sums of generalized permutohedra are generalized permutohedra (Propositions 2.14 and 2.16).
Used to write Newton(D_w) as a generalized permutohedron in Theorem 3.18.
standard math Products of nonnegative linear forms have SNP, and vertices of Minkowski sums of simplices are points with a unique representation (Agnarsson-Morris [1, Theorems 3.4 and 3.5]).
Supplies the vertex characterization in Corollary 1.4.
standard math Vertices of the relevant generalized permutohedra correspond to rectangle tilings of staircases (Postnikov [14, Corollary 8.2]).
Gives the tiling rule in Corollary 4.1.
standard math Bruhat cover property: if v ⋖ w = v t_ab, then no value among positions a+1,...,b-1 lies between w(a) and w(b).
Used implicitly in Lemma 3.9 to compare inversion sets; standard in Bruhat order theory.

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Cite this review

Pith. "Pith review of Newton polytopes of dual Schubert polynomials." pith.science (2026). https://pith.science/paper/S72AHT2E

@misc{pith2026241116654,
  author       = {Pith},
  title        = {Pith review of: Newton polytopes of dual Schubert polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S72AHT2E}},
  note         = {Machine review of arXiv:2411.16654}
}

read the original abstract

The M-convexity of dual Schubert polynomials was first proven by Huh, Matherne, M\'esz\'aros, and St. Dizier in 2022. We give a full characterization of the supports of dual Schubert polynomials, which yields an elementary alternative proof of the M-convexity result, and furthermore strengthens it by explicitly characterizing the vertices of their Newton polytopes combinatorially.

Figures

Figures reproduced from arXiv: 2411.16654 by the authors.

**Figure 1.** Calculating the Postnikov–Stanley polynomial D321 213. The support and Newton polytope of a polynomial behave nicely with respect to polynomial addition and multiplication as follows. Proposition 2.7. For two polynomials f, g ∈ R≥0[x1, . . . , xn], we have supp(fg) = supp(f) + supp(g) supp(f + g) = supp(f) ∪ supp(g), where + in the first line denotes the Minkowski sum. Proposition 2.8. ([12]) For two polynomials f,… view at source ↗

**Figure 2.** Newton(D254361) has a vertex at (0, 1, 0, 6, 1). Example 4.3 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Newton(D4213) has vertices (3, 1, 0),(1, 3, 0),(1, 2, 1),(2, 1, 1). 5. Future Directions We hope to generalize our results to Postnikov–Stanley polynomials. Conjecture 5.1. Postnikov–Stanley polynomials are M-convex [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

17 extracted references · 11 canonical work pages

[1]

On minkowski sums of simplices

Geir Agnarsson and Walter D Morris. On minkowski sums of simplices . Annals of Combinatorics , 13(3):271–287, 2009

work page 2009
[2]

Matroid p olytopes and their volumes

Federico Ardila, Carolina Benedetti, and Jeﬀrey Doker. Matroid p olytopes and their volumes. Discrete and Computational Geometry , 43, 10 2008

work page 2008
[3]

Schubert cells and cohomology of the spaces g/p

Bernstein, Gelfand, and Gelfand. Schubert cells and cohomology of the spaces g/p. Russian Mathemat- ical Surveys , 28:1–26, 1973

work page 1973
[4]

Combinatorics of Coxeter Groups

Anders Bjorner and Francesco Brenti. Combinatorics of Coxeter Groups . Springer Berlin, Heidelberg, 01 2005

2005
[5]

Double schubert polyn omials do have saturated newton polytopes

Castillo, Cid Ruiz, Mohammadi, and Montano. Double schubert polyn omials do have saturated newton polytopes. Forum Math Sigma , 11:e100, 2023

work page 2023
[6]

Double Schubert polynomials do have saturated Newton polytopes

Federico Castillo, Yairon Cid-Ruiz, Fatemeh Mohammadi, and Jonat han Monta˜ no. Double Schubert polynomials do have saturated Newton polytopes. Forum Math. Sigma , 11:Paper No. e100, 9, 2023

work page 2023
[7]

Fink, M´ ez´ saros, and St. Dizier. Schubert polynomials as integer point transforms of generalized permu- tahedra. Advances in Math , 332:465–475, 2018

2018
[8]

An involution on RC-graphs and a conjecture on dual Sc hubert polynomials by Postnikov and Stanley

Yibo Gao. An involution on RC-graphs and a conjecture on dual Sc hubert polynomials by Postnikov and Stanley. Algebr. Comb. , 3(3):593–602, 2020

work page 2020

Show all 17 references

[9]

Newton polytopes of the cla ssical resultant and discriminant

Gelfand, Kapranov, and Zelevinsky. Newton polytopes of the cla ssical resultant and discriminant. Ad- vances in Math , 84:237–254, 1990

1990
[10]

Dual schubert polynomials via a cauchy ident ity, 2023

Zachary Hamaker. Dual schubert polynomials via a cauchy ident ity, 2023

2023
[11]

Matherne, Karola M´ esz´ aros, and Avery St

June Huh, Jacob P. Matherne, Karola M´ esz´ aros, and Avery St. Dizier. Logarithmic concavity of Schur and related polynomials. Trans. Amer. Math. Soc. , 375(6):4411–4427, 2022

2022
[12]

Newton poly topes in algebraic combinatorics

Cara Monical, Neriman Tokcan, and Alexander Yong. Newton poly topes in algebraic combinatorics. Selecta Mathematica, 25, 2019

2019
[13]

Discrete Convex Analysis

Kazuo Murota. Discrete Convex Analysis . Society for Industrial and Applied Mathematics, 2003

2003
[14]

Permutohedra, associahedra, and bey ond

Alexander Postnikov. Permutohedra, associahedra, and bey ond. International Mathematics Research Notices, 2009, 08 2005

2009
[15]

Alexander Postnikov and Richard P. Stanley. Chains in the Bruha t order. J. Algebraic Combin. , 29(2):133–174, 2009

2009
[16]

An inequality

Richard Rado. An inequality. J. London Math. Soc. , 27:1–6, 1952

1952
[17]

SageMath, the Sage Mathematics Software System (Version 10 .4)

The Sage Developers. SageMath, the Sage Mathematics Software System (Version 10 .4). The Sage Development Team, 2024. Massachusetts Institute of Technology Email address : anser@mit.edu Harvard University Email address : katherinetung@college.harvard.edu University of Michigan...

2024

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

On minkowski sums of simplices

Geir Agnarsson and Walter D Morris. On minkowski sums of simplices . Annals of Combinatorics , 13(3):271–287, 2009

work page 2009

[2] [2]

Matroid p olytopes and their volumes

Federico Ardila, Carolina Benedetti, and Jeﬀrey Doker. Matroid p olytopes and their volumes. Discrete and Computational Geometry , 43, 10 2008

work page 2008

[3] [3]

Schubert cells and cohomology of the spaces g/p

Bernstein, Gelfand, and Gelfand. Schubert cells and cohomology of the spaces g/p. Russian Mathemat- ical Surveys , 28:1–26, 1973

work page 1973

[4] [4]

Combinatorics of Coxeter Groups

Anders Bjorner and Francesco Brenti. Combinatorics of Coxeter Groups . Springer Berlin, Heidelberg, 01 2005

2005

[5] [5]

Double schubert polyn omials do have saturated newton polytopes

Castillo, Cid Ruiz, Mohammadi, and Montano. Double schubert polyn omials do have saturated newton polytopes. Forum Math Sigma , 11:e100, 2023

work page 2023

[6] [6]

Double Schubert polynomials do have saturated Newton polytopes

Federico Castillo, Yairon Cid-Ruiz, Fatemeh Mohammadi, and Jonat han Monta˜ no. Double Schubert polynomials do have saturated Newton polytopes. Forum Math. Sigma , 11:Paper No. e100, 9, 2023

work page 2023

[7] [7]

Fink, M´ ez´ saros, and St. Dizier. Schubert polynomials as integer point transforms of generalized permu- tahedra. Advances in Math , 332:465–475, 2018

2018

[8] [8]

An involution on RC-graphs and a conjecture on dual Sc hubert polynomials by Postnikov and Stanley

Yibo Gao. An involution on RC-graphs and a conjecture on dual Sc hubert polynomials by Postnikov and Stanley. Algebr. Comb. , 3(3):593–602, 2020

work page 2020