REVIEW 3 major objections 4 minor 1 cited by
Newton polytopes of fireworks Grothendieck polynomials
T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read For fireworks permutations, a Grothendieck monomial appears exactly when it lies between a Schubert monomial and the top-degree bound.
desk verdict The support formula for fireworks Grothendieck polynomials is new and likely correct, but the proof of the key raising lemma has two unproved existence assertions that a referee should push on. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine of the proof is a local surgery on pipe dreams, which are tilings of a staircase grid by cross tiles and bump tiles that encode the monomials of Schubert and Grothendieck polynomials. Theorem 1.3 states that for a fireworks permutation, whenever a pipe dream's weight vector lies strictly below the maximal weight $\mathrm{wt}(\overline{D(w)})$ in some coordinate $a$, there is another pipe dream for the same permutation whose first $a-1$ weights agree, whose $a$-th weight is one larger, and whose later weights are no larger. The surgery replaces a carefully chosen bump tile in row $a$ by a cross tile, then repairs the pipe network below by flipping certain crossings to bumps; Lemmas 3.1, 3.2, and 3.3 control which pipes can pass between two boundary tiles so that the permutation is unchanged. Iterating this raising operation from reduced pipe dreams, which give the Schubert monomials, fills every lattice point between a Schubert exponent and the maximal vector, which is exactly the support formula.
What would settle it
Compute the Grothendieck polynomial $\mathfrak{G}_w$ for every fireworks permutation of small size, for example $n \le 6$, and compare $\mathrm{supp}(\mathfrak{G}_w)$ with the union over $\alpha \in \mathrm{supp}(\mathfrak{S}_w)$ of $[\alpha, \mathrm{wt}(\overline{D(w)})]$. If any monomial that divides $\mathbf{x}^{\mathrm{wt}(\overline{D(w)})}$ and is divisible by some Schubert monomial has coefficient zero in $\mathfrak{G}_w$, the theorem is false; even one minimal counterexample would settle it.
Extended reading notes
Core claim
On the paper's own terms, the central result is Theorem 1.1: if $w$ is fireworks, then $\mathrm{supp}(\mathfrak{G}_w)$ equals the union over all Schubert monomial exponents $\alpha \in \mathrm{supp}(\mathfrak{S}_w)$ of the componentwise intervals $[\alpha, \mathrm{wt}(\overline{D(w)})]$, where $\overline{D(w)}$ is the upward closure of the Rothe diagram. Equivalently, every monomial that divides $\mathbf{x}^{\mathrm{wt}(\overline{D(w)})}$ and is divisible by some monomial of $\mathfrak{S}_w$ occurs with nonzero coefficient in $\mathfrak{G}_w$. The proof builds, for any pipe dream of weight below the maximum, another pipe dream of the same permutation that increases one chosen row's crossing count by one without increasing later rows; iterating this surgery fills out the whole interval. From this formula the paper derives that the homogenized Grothendieck polynomial has M-convex support, hence a saturated Newton polytope and a generalized-permutahedron Newton polytope, and that the support is contained in that of the layered permutation whose block sizes are the lengths of the descending runs of $w$.
Load-bearing premise
The whole interval description rests on the geometric fact that in a fireworks pipe dream, whenever a row has fewer crossings than the maximum, a suitable bump tile exists in that row whose primary pipe is not a left-to-right maximum; if that local tile search ever fails, the weight-raising construction stops and Theorem 1.1 does not follow.
Editorial extensions
If this is right
- For fireworks $w$, membership in $\mathrm{supp}(\mathfrak{G}_w)$ is decidable by two divisibility checks: $\mathbf{x}^\alpha$ divides $\mathbf{x}^{\mathrm{wt}(\overline{D(w)})}$ and some Schubert monomial divides $\mathbf{x}^\alpha$.
- The Newton polytope of $\mathfrak{G}_w$ equals the Minkowski sum of Schubert spanning set polytopes attached to the columns of the Rothe diagram, intersected with $\mathbb{Z}^n$; hence it is computable column by column.
- The homogenized Grothendieck polynomial of a fireworks permutation has M-convex support, so its Newton polytope is a generalized permutahedron and the polynomial has the saturated Newton polytope property.
- For every fireworks $w$, the layered permutation $\pi(w)$ with block sizes equal to the descending runs of $w$ satisfies $\mathrm{supp}(\mathfrak{G}_{\pi(w)}) \supseteq \mathrm{supp}(\mathfrak{G}_w)$.
- The maximal monomial $\mathbf{x}^{\mathrm{wt}(\overline{D(w)})}$ appears in $\mathfrak{G}_w$ for fireworks $w$, recovering a known existence statement as a byproduct of the construction.
Reading between the lines
- If the same interval-union principle holds beyond fireworks, it would give a uniform explanation of the known M-convexity cases; the natural test case is a permutation whose top-degree Grothendieck part is a single monomial, where the inclusion in Remark 4.5 would become an equality.
- The column-by-column Minkowski description suggests an efficient membership algorithm: check the Schubert support via matroid polytopes and then check the upper bound; the paper does not discuss complexity, but the structure is algorithmically friendly.
- The layered containment may be strict in general, and comparing it with known extremal behavior of layered permutations in Schubert calculus suggests that fireworks Grothendieck polynomials are also extremal for support size, a statement the paper does not make.
- One could test the sharpness of the fireworks hypothesis by searching for a non-fireworks permutation for which the union of intervals differs from the actual support; a positive example would delineate exactly why the pipe-dream surgery needs the 3-12 avoidance condition.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the Newton polytopes of Grothendieck polynomials for fireworks permutations. Its main result, Theorem 1.1, asserts that the support of such a Grothendieck polynomial is as large as possible subject to known bounds: a monomial appears exactly when it divides x^{wt(\overline{D(w)})} and is divisible by some monomial in the corresponding Schubert polynomial. The proof is built on a pipe-dream raising operation (Theorem 1.3) that increases the weight of a fixed row without changing the permutation. From the support formula the authors derive two corollaries: the homogenized Grothendieck polynomial has M-convex support, and for every fireworks permutation w there is a layered permutation whose Grothendieck polynomial has support containing that of w.
Significance. If the central theorem is correct, it gives a complete support description for a substantial family of Grothendieck polynomials, extending the known SNP and M-convexity results for Schubert polynomials and for vexillary Grothendieck polynomials. The resulting M-convexity of the homogenized support and the layered-permutation containment are natural and interesting consequences. The paper works with explicit pipe dreams and elementary polytopal tools, so the intended audience is broad. The main risk is that the proof of the key raising lemma, Theorem 1.3, relies on unproved local assertions about pipe-dream topology; these assertions are load-bearing for the main theorem.
major comments (3)
- [Section 3, proof of Theorem 1.3, first sentence] The proof begins with the assertion that if wt(P)_a < wt(\overline{D(w)})_a, then there exists a primary pipe P(i) of a bump tile T in row a such that i is not a left-to-right maximum. This is load-bearing: the entire raising construction starts from such a T. The assertion is not an immediate consequence of Proposition 2.10, which counts non-initial-term positions j>a, not primary pipes of bump tiles in row a. A simple counting check shows the gap: for the fireworks permutation w=31542 and a=3, wt(\overline{D(w)})_3=2, while the number of non-left-to-right-maximum pipes with exit row greater than 3 is 1. Thus the deficit in the bound does not automatically produce the claimed bump tile. The authors should supply a rigorous counting or geometric argument, or state and prove the precise lemma they are using.
- [Section 3, Case 2 of proof of Theorem 1.3] The case analysis asserts that 'The pipe P(ℓ) is primary for some tile S in row a' and then, using Lemma 3.1(1), that S is a bump tile. The first assertion is plausible because P(ℓ) has exit row greater than a, but it is not stated or proved; the second should be a formal application of Lemma 3.1 rather than an implicit step. In addition, the later reference to P(ℓ') is undefined: the secondary pipe of S was earlier called P(m), and the text then uses P(ℓ') without introducing it. The authors should rewrite this case so that every pipe label is defined and every use of Lemmas 3.1-3.3 is explicit.
- [Section 1, proof of Theorem 1.1] The sentence 'By repeated application of Theorem 1.3, there exists a pipe dream Q with x^{wt(Q)} = x_i x^\alpha' is too terse, because Theorem 1.3 allows weights in rows b>a to decrease. To reach the exact vector \alpha+ e_i, one must apply the theorem first to coordinate i and then restore rows n, n-1, ..., i+1 from right to left, since increasing a row b>i does not affect any row to its left. This ordering should be stated. Without it, the reader cannot verify that the desired monomial is obtained without further changes.
minor comments (4)
- [Section 1, Introduction] The sentence 'Every monomial appearing in any Grothendieck polynomial G_w divides x^{wt(\overline{D(w)})}' is used in the statement of Theorem 1.1 but is neither proved nor cited. Please add a reference to a standard source or a short proof.
- [Section 3, Lemma 3.3] The statement contains grammatical and omission errors: 'the two pipes involved tiles T and T\'' should be 'involved in tiles T and T\'', and 'let denote the region R' should be 'let R denote the region'.
- [Section 3, Case 2 of proof of Theorem 1.3] The symbol P(ℓ') is used in the bulleted construction and in Figure 8 but is never defined. It should either be replaced by P(m), the secondary pipe of S, or defined explicitly before use.
- [Section 4, Proposition 4.4] In the displayed formula, 'Psp(SMdj (Dj))' is a typo; it should read Psp(SM(D_j)).
Circularity Check
No significant circularity: the support formula is derived from a self-contained pipe-dream raising construction, with self-citations only motivational or re-proved.
full rationale
The central derivation is not circular. Theorem 1.1 is obtained by iterating Theorem 1.3, whose proof is a local pipe-dream surgery based on Lemmas 3.1-3.3 that are proved in the text. The union-of-intervals support formula is then combined with the external FMS18 description of Schubert support (Theorem 2.17) and the standard pipe-dream expansions of Schubert and Grothendieck polynomials (Theorem 2.2). The MSS25 conjecture appears only as motivation, and the main claim is not assumed from it. The cited self-works are not load-bearing: [MSS25] is cited for the motivational conjecture and for Lemma 4.4, whose proof is restated in the paper; [CY25] is cited for Lemma 3.3, but the lemma is also proved here; [HMSS24] and [PSW24] are context citations. The geometric assertions in the opening of the proof of Theorem 1.3 are gaps or sketch points if one demands more detail, but they are not reductions of the conclusion to its own input. There is no fitted parameter renamed as a prediction and no uniqueness claim imported from the authors' prior work. The derivation chain, as written, is self-contained against standard external results.
Assumptions & free parameters
assumptions (6)
- standard math Pipe dream expansions of Schubert and Grothendieck polynomials (Theorem 2.2, citing FK94, KM04, Wei21).
- standard math Schubert polynomial supports are M-convex and decompose as Minkowski sums of Schubert matroid polytopes (Theorem 2.17, citing FMS18).
- standard math Integer points in a Minkowski sum of generalized permutahedra are sums of integer points of summands (Lemma 2.15, citing Sch03).
- standard math Homogenized Schubert spanning sets form an M-convex set, after Fujishige (Definition 4.2).
- domain assumption Combinatorial characterization of fireworks permutations via decreasing runs (Proposition 2.10 and Remark 2.12).
- standard math Every monomial in any Grothendieck polynomial divides x^{wt(D(w))}, and for fireworks the top-degree part is a scalar multiple of x^{wt(D(w))} (stated in the introduction, citing MSS25 and CY25).
Cite this review
Pith. "Pith review of Newton polytopes of fireworks Grothendieck polynomials." pith.science (2026). https://pith.science/paper/WBBATPQP
@misc{pith2026250809107,
author = {Pith},
title = {Pith review of: Newton polytopes of fireworks Grothendieck polynomials},
year = {2026},
howpublished = {\url{https://pith.science/paper/WBBATPQP}},
note = {Machine review of arXiv:2508.09107}
}
abstract
We show that the support of the Grothendieck polynomial $\mathfrak G_w$ of any fireworks permutation is as large as possible: a monomial appears in $\mathfrak G_w$ if and only if it divides $\mathbf x^{\mathrm{wt}(\overline{D(w)})}$ and is divisible by some monomial appearing in the Schubert polynomial $\mathfrak S_w$. Our formula implies that the homogenization of $\mathfrak G_w$ has M-convex support. We also show that for any fireworks permutation $w\in S_n$, there exists a layered permutation $\pi(w)\in S_n$ so that $\mathrm{supp}(\mathfrak G_{\pi(w)})\supseteq \mathrm{supp}(\mathfrak G_w)$.
Forward citations
Cited by 1 Pith paper
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Asymptotically maximal Schubitopes
The maximal support size of Schubert polynomials is asymptotically between n!/4^n and n!, and for Grothendieck polynomials it is n! times a subexponential factor.
Reference graph
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Reviewed August 15, 2026 · model on record in the stance chip above.
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