REVIEW 3 major objections 3 minor 1 references
Asymptotically maximal Schubitopes
T0 review · 3 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash
Pith's one-line read This paper proves the maximal support size of a Schubert polynomial grows like n^n up to a subexponential factor, and pins the second-order Grothendieck growth exactly at -1.
desk verdict Solid self-contained Schubert half, sharp Grothendieck half that leans on an unproved companion formula; referee it. 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 main tool is the diagrammatic description of Schubert support: a vector α is in supp(S_w) iff α is the weight of a diagram C lying weakly northwest of the Rothe diagram D(w) (C ≤ D(w)). For layered permutations, D(w) splits into blocks, and the paper proves (Lemma 3.2) a fine subset-replacement property: inside the last block, any chosen subset of boxes can be realized by some C ≤ D(w) without changing its weight elsewhere. This lets the authors embed b_m! disjoint translated copies of the smaller Schubitope S_{D(w')} into S_{D(w)}, giving the recursion |supp(S_w)| ≥ b_m! |supp(S_w')|. For Grothendieck polynomials, a formula from a companion paper for fireworks permutations expresses sup
What would settle it
Verify Proposition 4.1 on a small non-layered fireworks permutation, e.g. w=2413, by brute-force computing supp(G_w) and checking it equals the claimed union of intervals [α, wt(overline(D(w)))]; if any monomial lies outside the intervals, the exact Grothendieck asymptotics collapse. Also check the recursion in Theorem 3.1 on a concrete layered permutation at, say, n=10 by enumerating |supp(S_w)| and comparing it to the product-of-factorials lower bound.
Extended reading notes
Core claim
The central discovery is that layered permutations — permutations made of decreasing blocks — are large-support extremizers. Theorem 1.1 establishes that for β(n) = max_{w∈S_n} |supp(S_w)|, one has ln β(n)/(n ln n) → 1, and more precisely -ln4-1 ≤ liminf (ln β(n)-n ln n)/n ≤ limsup (ln β(n)-n ln n)/n ≤ -1. Theorem 1.2 proves the analogous quantity for Grothendieck polynomials satisfies lim (ln β^G(n)-n ln n)/n = -1 exactly. The construction gives explicit layered permutations achieving the stated lower bounds.
Load-bearing premise
The exact Grothendieck result rests on a support-interval formula from a companion paper that is not proved here; if that formula is wrong, Theorem 1.2 fails.
Editorial extensions
If this is right
- The maximal Schubert support satisfies β(n) = n^{n - o(1)} (equivalently ln β(n) ~ n ln n), so the first-order growth is now known exactly.
- There exist explicit layered permutations with support at least n!/4^n; these provide concrete candidates for support-maximizers.
- The maximal Grothendieck support has the exact second-order asymptotics ln β^G(n) = n ln n - n + O(√n ln n), leaving only a sub-√n correction undetermined.
- The recursive embedding proves a stronger statement: the Schubitope of a layered permutation contains many disjoint integer translates of the Schubitope of its prefix, giving a structural insight into support saturation.
- These results settle a problem posed in earlier literature on the growth rate of maximal Schubert support.
Reading between the lines
- The recursive block construction suggests a general principle: any permutation whose Rothe diagram splits into two separated parts will have support at least the product of the supports of its components; this might extend the result from layered permutations to other diagram families.
- The gap between the lower bound -ln4-1 and the upper bound -1 in Theorem 1.1 leaves open the possibility that the true second-order constant is -1; if so, a construction avoiding the 4^n loss in the product-of-factorials bound would close the gap.
- For Grothendieck polynomials, the interval formula implies the support is the integer points of a box [c,d]; maximizing |supp(G_w)| is then a purely combinatorial question about Lehmer codes, which could be attacked by optimization over diagrams rather than permutations.
- The same layered permutations might be tested as extremizers for the principal specialization S_w(1,...,1), connecting the support-maximization problem to the older degree-maximization problem.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies β(n), the maximum size of the support of a Schubert polynomial over permutations in S_n, and the analogous quantity β^G(n) for Grothendieck polynomials. The main result (Theorem 1.1) asserts that ln β(n)/(n ln n) → 1, with explicit bounds -ln 4 - 1 ≤ liminf (ln β(n)-n ln n)/n ≤ limsup (...) ≤ -1. The proof constructs layered permutations whose Schubert support is at least ∏ floor(n/2^k)!, using a recursive embedding into Schubitopes (Theorem 3.1 and Corollary 3.3), and matches this with the trivial upper bound n!. For Grothendieck polynomials, Theorem 1.2 asserts ln β^G(n) = n ln n - n + o(n), equivalently (ln β^G(n)-n ln n)/n → -1. The lower bound uses a fireworks-permutation support formula from the authors' companion paper [CS25]. Both lower-bound constructions are explicit and are layered/fireworks permutations.
Significance. If correct, Theorem 1.1 completely resolves the first-order asymptotics of the maximal Schubert support, answering a problem of Guo–Lin [GL24, Prob 5.5], and gives the first explicit bounds on the second-order term. The proof is elementary, self-contained (for the Schubert part), and has no fitted parameters or reverse-engineered constants. Theorem 1.2 gives a similarly sharp asymptotic for Grothendieck supports, though its lower bound depends on an external support formula for fireworks permutations. The paper is a solid, incremental advance in the asymptotic combinatorics of Schubert polynomials, with a clean main construction that should be of interest to the field.
major comments (3)
- [§3, Theorem 3.1 and Lemma 3.2] The proof of Theorem 3.1 states: 'use Lemma 3.2 to choose a diagram C_α ≤ D(w_LB) whose weight satisfies wt(C_α)_{n-b_m+j} = α_j.' But Lemma 3.2 only guarantees the existence of C with C ∩ D(w_LB) = S; it does not assert any control on wt(C). In general C∩D=S does not imply that the row-weight vector of C agrees with that of S, because C may have boxes outside D. The proof as written is therefore incomplete at a load-bearing point. The gap is fixable: in the proof of Lemma 3.2, the extra boxes are placed below the minimum row of D_j, i.e., in rows ≤ n-b_m, and D(w_LB) has all its boxes in rows n-b_m+1,...,n; one should state explicitly that the added boxes do not affect the last b_m coordinates of wt(C). I recommend strengthening Lemma 3.2 (or adding a sentence in the proof of Theorem 3.1) to record this weight-control property.
- [§4, Proposition 4.1 and Theorem 1.2] The entire lower bound for Grothendieck support, hence the exact coefficient -1 in Theorem 1.2, depends on Proposition 4.1, which is quoted from the companion paper [CS25, Thm 1.1] without proof. Since [CS25] is a separate preprint by the same authors, a reader of this manuscript alone cannot certify Theorem 1.2. This is standard citation practice in one sense, but because Proposition 4.1 is the only input that upgrades the lower bound from qualitative to quantitative, the manuscript should either (a) include a proof or a detailed proof sketch of Proposition 4.1, or (b) explicitly state in the introduction and in Theorem 1.2 that the result is conditional on [CS25]. As written, the statement of Theorem 1.2 is not self-contained.
- [§4, Proposition 4.2] The product lower bound in the proof of Proposition 4.2 is not valid for all n. For example, when n=3, the unique integer k is 2, and the product over j=1 to k-1 of (n-k-1 - sum_{i=1}^{j-1} i)^j evaluates to 0^1 = 0, while (n-k-1)! = 0! = 1. Thus the displayed chain 'product ≥ (n-k-1)! ≥ n!/n^{k+1}' fails for n=3. This does not affect the asymptotic statement, since for n ≥ 4 the inequality appears to hold, but the proof should either handle small n separately or state that n is assumed sufficiently large.
minor comments (3)
- [§2, Lemma 2.11] The statement of Lemma 2.11 appears to be \(\lfloor k\rfloor! \ge \frac{1}{2^k}(k/e)^k\), but the typeset inequality in the text could be misread as \(\frac{1}{2k}\). Please clarify the notation.
- [§3, Corollary 3.3] In the displayed chain of inequalities, the step involving \(1/n^c\) and the factor \(2^{-2n}\) is compressed; adding one line showing the product of the \((1/2^k)^{n/2^k}\) factors is at least \(2^{-2n}\) would improve readability.
- [References] The reference [GL24] is cited as arXiv:2412.02932; if a published version exists, it would be good to update the citation.
Circularity Check
No circularity found; Theorem 1.1 is self-contained and Theorem 1.2 relies on a prior companion-paper theorem, not on a restatement of this paper's conclusions.
full rationale
The main internal contribution, Theorem 1.1, is derived directly from Lemma 3.2, Theorem 3.1, and Corollary 3.3. The support lower bound is constructed by explicit disjoint translations indexed by vectors chosen from n, not from the target asymptotics, and the upper bound uses Lemma 2.5 plus factorial estimates. No fitted parameter is renamed as a prediction, and no definition is circular. For Theorem 1.2, the lower bound does import Proposition 4.1 from the authors' companion paper [CS25, Thm 1.1], giving the interval [wt(D(w)), wt(overline D(w))] inside the Grothendieck support of fireworks permutations. This citation is load-bearing for the exact -1 coefficient in Theorem 1.2, and a reader of this preprint alone cannot independently audit the companion proof. However, that is ordinary citation of prior work, not a reduction of the present claim to its own definition or fit. The companion theorem is not asserted as a consequence of this paper's Theorem 1.2; rather, Theorem 1.2 uses it as an external input. The paper itself does not claim to prove Proposition 4.1 here, and no self-referential uniqueness or ansatz-via-citation pattern appears. There is also a small non-circular arithmetic inconsistency in Proposition 4.2 at n=3, but it concerns correctness, not circularity. Therefore no step qualifies as circular under the stated criteria; score 0.
Assumptions & free parameters
assumptions (3)
- domain assumption Schubert support equals the set of weights of subdiagrams C ≤ overline D(w), per FMS18, Thm 4.
- domain assumption Fireworks Grothendieck support formula of [CS25, Thm 1.1].
- domain assumption Monomials in a Grothendieck polynomial divide the monomial x^{D(w)}, per MSS22, Thm 1.2.
Cite this review
Pith. "Pith review of Asymptotically maximal Schubitopes." pith.science (2026). https://pith.science/paper/C2VSN3QJ
@misc{pith2026251204053,
author = {Pith},
title = {Pith review of: Asymptotically maximal Schubitopes},
year = {2026},
howpublished = {\url{https://pith.science/paper/C2VSN3QJ}},
note = {Machine review of arXiv:2512.04053}
}
abstract
We find a layered permutation $w\in S_n$ whose Schubert polynomial $\mathfrak S_w(x_1, \dots, x_n)$ has support of size asymptotically at least $n!/4^n$. This gives precise asymptotics for the growth rate of $\beta(n):= \max_{w\in S_n}|\mathrm{supp}(\mathfrak S_w)|$. We find a different layered permutation $w\in S_n$ whose Grothendieck polynomial has support of size asymptotically at least $n!/e^{\sqrt{2n} \cdot \ln(n)}$ and obtain more precise asymptotics for the growth rate of $\beta^{\mathfrak G}(n):=\max_{w\in S_n}|\mathrm{supp}(\mathfrak G_w)|$.
Reference graph
Works this paper leans on
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[1]
[CS25] Jack C. A. Chou and Linus Setiabrata,Newton polytopes of fireworks Grothendieck polynomials(2025), available at arXiv:2508.09107. [FMS18] Alex Fink, Karola M ´esz´aros, and Avery St. Dizier,Schubert polynomials as integer point transforms of generalized per- mutahedra, Adv. Math.332(2018), 465–475. [Gao21] Yibo Gao,Principal specializations of Schu...
arXiv 2025
Reviewed August 3, 2026 · model on record in the stance chip above.
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