pith. machine review for the scientific record. sign in

arxiv: 2605.10016 · v1 · submitted 2026-05-11 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

Lattice-free Schubitopes

Jinren Dou, Kunwen Liu, Neil J.Y. Fan

Pith reviewed 2026-05-12 03:01 UTC · model grok-4.3

classification 🧮 math.CO
keywords Schubitopeslattice-free polytopesEhrhart polynomialsSchubert polynomialsGrothendieck polynomialsNewton polytopespattern avoidance
0
0 comments X

The pith

A Schubitope is lattice-free if and only if its Ehrhart polynomial is the product of the Ehrhart polynomials of its column Schubert matroid polytopes.

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

The paper supplies a simple criterion for deciding whether the Schubitope attached to a given diagram is lattice-free. It proves this property holds if and only if the Ehrhart polynomial of the Schubitope is exactly the product of the Ehrhart polynomials coming from the Schubert matroid polytopes of each column. The criterion yields that the Newton polytopes for Schubert polynomials and Grothendieck polynomials are lattice-free precisely for permutations avoiding the patterns 1423, 1432, and 13254, thereby settling several open conjectures about the supports of the Grothendieck polynomials in these cases.

Core claim

We provide a simple criterion for the Schubitope S_D associated to a diagram D to be lattice-free. We further show that S_D is lattice-free if and only if its Ehrhart polynomial is equal to the product of Ehrhart polynomials of the Schubert matroid polytopes corresponding to each column of D. As applications, we obtain that the Newton polytopes of the Schubert polynomial and the Grothendieck polynomial are lattice-free if and only if the permutation avoids the patterns 1423, 1432, 13254, and we confirm several conjectures on the support of Grothendieck polynomials for this class of permutations.

What carries the argument

The Schubitope S_D of a diagram D, characterized for lattice-freeness by whether its Ehrhart polynomial equals the product of the Ehrhart polynomials of the Schubert matroid polytopes for each column of D.

If this is right

  • The Newton polytope of the Schubert polynomial is lattice-free precisely when the permutation avoids 1423, 1432, and 13254.
  • The Newton polytope of the Grothendieck polynomial is lattice-free for the same avoiding permutations.
  • Conjectures regarding the support of Grothendieck polynomials are confirmed for pattern-avoiding permutations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The factorization condition may simplify volume calculations or other geometric invariants for these polytopes.
  • This links pattern avoidance directly to the geometry of lattice points in Newton polytopes.
  • The criterion could be tested on other diagram-based constructions to find more lattice-free examples.

Load-bearing premise

The standard definitions of Schubitopes, Schubert matroid polytopes, and their Ehrhart polynomials apply to the diagrams D, and the pattern-avoidance conditions correctly identify the lattice-free Newton polytopes.

What would settle it

A diagram D for which the Schubitope has no interior lattice points but its Ehrhart polynomial does not match the product of the column Schubert matroid Ehrhart polynomials.

Figures

Figures reproduced from arXiv: 2605.10016 by Jinren Dou, Kunwen Liu, Neil J.Y. Fan.

Figure 1.1
Figure 1.1. Figure 1.1: D = ({1, 3}, {2, 3}, {1}) Theorem 1.1. For any diagram D = (D1, . . . , Dn) ⊆ [n] 2 , the following conditions are equivalent: (1) The Schubitope SD is lattice-free; (2) The movable intervals in different columns of D intersect in at most one element, i.e., for any 1 ≤ i < j ≤ n, |M(Di) ∩ M(Dj )| ≤ 1; (3) The Ehrhart polynomial of the Schubitope SD factors as i(SD, t) = Yn j=1 i(P(SMn(Dj )), t), (1.1) wh… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: A Rothe diagram and a skyline diagram 2.2. Schubitopes. Let D = (D1, . . . , Dn) ⊆ [n] 2 be an arbitrary diagram. Let I ⊆ [n] be a set of row indices and j ∈ [n] be a column index. Construct the string wordI (Dj ) in the following way. Read the j-th column of the n × n grid from top to bottom and record: (1) If i /∈ Dj and i ∈ I, record a left parenthesis (; (2) If i ∈ Dj and i /∈ I, record a right paren… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: D = {D1, D2, . . . , Dk} Suppose that v1, v2, . . . , vl are any l distinct vertices in Pk j=1 P(SMn(Dj )). If the convex hull of v1, v2, . . . , vl contains a lattice point v ′ , and v ′ ̸= v1, . . . , vl , then v ′ can be written as v ′ = λ1v1 + λ2v2 + · · · + λlvl , where 0 < λi < 1 and λ1 + λ2 + · · · + λl = 1. We will show that v ′ = v1 = v2 = · · · = vl , leading to a contradiction. By our assumpti… view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: The illustration of Case 1 Case 2. B is not in the same row of A. Then B must locate in a lower row of A. Let B = (i2, b) with i2 > i1 and b > a. Then there is a hook with corner at (i1, w(i1)) with b > w(i1) > a. Now consider the position at (i2, a). If there is a box at (i2, a), then delete row i1 and the column w(i1), the discussion returns to Case 1, see the left figure in [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: The illustration of Case 2 5.2. Newton polytopes of Grothendieck polynomials. Recently, M´esz´aros, Se￾tiabrata, and St. Dizier [27] conducted a systematic study on the support supp(Gw) of Grothendieck polynomials. Equipping supp(Gw) ⊂ Z n with the componentwise compari￾son order α ≤ β, they proposed several conjectures (Conjectures 1.1–1.6 in [27]) detailing the combinatorial structure and saturatedness… view at source ↗
read the original abstract

In this paper, we provide a simple criterion for the Schubitope $\mathcal{S}_{D}$ associated to a diagram $D$ to be lattice-free. We further show that $\mathcal{S}_{D}$ is lattice-free if and only if its Ehrhart polynomial is equal to the product of Ehrhart polynomials of the Schubert matroid polytopes corresponding to each column of $D$. As applications, we obtain that the Newton polytopes of the Schubert polynomial $\mathfrak{S}_w(x)$ and the Grothendieck polynomial $\mathfrak{G}_w(x)$ are lattice-free if and only if $w$ avoids the patterns 1423, 1432, 13254, and confirm several conjectures by M\'esz\'aros, Setiabrata, and St.Dizier on the support of Grothendieck polynomials for this class of permutations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript provides a criterion for when the Schubitope S_D associated to a diagram D is lattice-free. It proves that S_D is lattice-free if and only if its Ehrhart polynomial factors as the product of the Ehrhart polynomials of the Schubert matroid polytopes corresponding to the columns of D. As applications, it characterizes the lattice-freeness of the Newton polytopes of the Schubert polynomial S_w(x) and Grothendieck polynomial G_w(x) in terms of w avoiding the patterns 1423, 1432, and 13254, and confirms several conjectures of Mészáros, Setiabrata, and St.Dizier on the support of Grothendieck polynomials for this class of permutations.

Significance. If the central equivalence holds, the work supplies a concrete, checkable criterion for lattice-freeness of Schubitopes that directly ties to Ehrhart polynomial factorization, which is valuable for both theoretical and computational purposes in combinatorial polytope theory. The pattern-avoidance characterization for Newton polytopes of Schubert and Grothendieck polynomials resolves open questions and confirms prior conjectures, strengthening links between permutation patterns and polytope geometry.

minor comments (2)
  1. The introduction would benefit from a brief recall of the precise definition of a diagram D and the associated Schubitope S_D (currently referenced to prior literature) to improve self-containedness for readers outside the immediate subfield.
  2. In the statement of the main equivalence (presumably Theorem 1 or equivalent), the precise indexing of columns and the normalization of the Ehrhart polynomials should be made explicit to avoid any ambiguity in the product formula.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript, their assessment of its significance, and the recommendation of minor revision. The report does not list any specific major comments, so we have no points requiring point-by-point rebuttal or changes to the manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives a lattice-freeness criterion for Schubitopes S_D and an if-and-only-if equivalence between this property and the factorization of the Ehrhart polynomial of S_D into a product over column-wise Schubert matroid polytopes. These statements rest on the standard definitions of Schubitopes, Schubert matroid polytopes, and Ehrhart polynomials taken from the existing combinatorial literature, together with explicit pattern-avoidance conditions that are checked directly against the Newton polytopes of Schubert and Grothendieck polynomials. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional renaming; the central claims therefore remain independent of the paper's own inputs and are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters or invented entities can be identified; the work relies on established definitions and properties of Ehrhart polynomials and matroid polytopes.

axioms (2)
  • standard math Standard properties of Ehrhart polynomials for lattice polytopes
    The equivalence statement invokes Ehrhart polynomials, which are known to be well-defined for the polytopes in question.
  • domain assumption Definitions of Schubitopes and Schubert matroid polytopes from prior literature
    The paper applies these objects to diagrams D without re-deriving their basic properties.

pith-pipeline@v0.9.0 · 5439 in / 1412 out tokens · 54907 ms · 2026-05-12T03:01:23.026355+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Arun and T

    S. Arun and T. Dillon, Improved Helly numbers of product sets, arXiv:2409.07262. 1

    work page arXiv

  2. [2]

    B´ ar´ any and J.-M

    I. B´ ar´ any and J.-M. Kantor, On the number of lattice free polytopes, European J. Combin. 21(2000), no. 1, 103-110; MR1737330 1

    work page 2000

  3. [3]

    Castillo, Y

    F. Castillo, Y. Cid-Ruiz, F. Mohammadi, and J. Monta˜ no, K-polynomials of multiplicity-free vari- eties. arXiv:2212.13091 5.5

    work page arXiv

  4. [4]

    Demazure, Une nouvelle formule des caract` eres, Bull

    M. Demazure, Une nouvelle formule des caract` eres, Bull. Sci. Math. (2) 98(1974), no. 3, 163-172; MR0430001 1

    work page 1974

  5. [5]

    Demazure, Key polynomials and a flagged Littlewood–Richardson rule, Bull

    M. Demazure, Key polynomials and a flagged Littlewood–Richardson rule, Bull. Sci. Math. 98 (1974), 163-172. 2.1

    work page 1974

  6. [6]

    Deza and S

    M.-M. Deza and S. Onn, Lattice-free polytopes and their diameter, Discrete Comput. Geom. 13(1995), no. 1, 59-75; MR1300509 1

    work page 1995

  7. [7]

    Ehrhart, Sur les poly` edres rationnels homoth´ etiques ` andimensions, C

    E. Ehrhart, Sur les poly` edres rationnels homoth´ etiques ` andimensions, C. R. Acad. Sci. Paris 254(1962), 616-618; MR0130860 1

    work page 1962

  8. [8]

    N. J. Y. Fan and P. L. Guo, Vertices of Schubitopes, J. Combin. Theory, Ser. A 177(2021), Paper No. 105311, 20 pp.; MR4139110 5.7

    work page 2021

  9. [9]

    N. J. Y. Fan and P. L. Guo, Set-valued Rothe tableaux and Grothendieck polynomials. Adv. Appl. Math., 128(2021): 102203. 5.2

    work page 2021

  10. [10]

    N. J. Y. Fan and P. L. Guo, Upper bounds of Schubert polynomials, Sci. China Math. 65(2022), no. 6, 1319-1330; MR4429393 2.2

    work page 2022

  11. [11]

    N. J. Y. Fan, P. L. Guo, S. C. Y. Peng, and S. C. C. Sun, Lattice points in the Newton polytopes of key polynomials, SIAM J. Discrete Math. 34(2020), no. 2, 1281-1289; MR4109582 5.7

    work page 2020

  12. [12]

    A. Fink, K. M´ esz´ aros, and A. St. Dizier, Schubert polynomials as integer point transforms of gen- eralized permutahedra, Adv. Math. 332(2018), 465-475. 1, 2.2, 2.2

    work page 2018

  13. [13]

    A. Fink, K. M´ esz´ aros and A. St. Dizier, Zero-one Schubert polynomials, Math. Z. 297(2021), no. 3-4, 1023-1042; MR4229590 5.1, 5.5

    work page 2021

  14. [14]

    Fomin and A

    S. Fomin and A. N. Kirillov, Grothendieck polynomials and the Yang-Baxter equation, in Formal power series and algebraic combinatorics/S´ eries formelles et combinatoire alg´ ebrique, DIMACS, Piscataway, NJ, sd, pp. 183–189. 5.2

    work page

  15. [15]

    A. J. Hoffman, Binding constraints and Helly numbers, inSecond International Conference on Combinatorial Mathematics (New York, 1978), pp. 284–288, Ann. New York Acad. Sci., 319, New York Acad. Sci., New York. MR0556036 1

    work page 1978

  16. [16]

    Kannan and L

    R. Kannan and L. Lov´ asz, Covering minima and lattice-point-free convex bodies, Ann. of Math. (2) 128(1988), no. 3, 577-602; MR0970611. 1

    work page 1988

  17. [17]

    Kantor, On the width of lattice-free simplices, Compositio Math

    J.-M. Kantor, On the width of lattice-free simplices, Compositio Math. 118(1999), no. 3, 235-241; MR1711323 1

    work page 1999

  18. [18]

    Kantor, Triangulations of integral polytopes and Ehrhart polynomials, Beitr¨ age Algebra Geom

    J.-M. Kantor, Triangulations of integral polytopes and Ehrhart polynomials, Beitr¨ age Algebra Geom. 39(1998), no. 1, 205-218; MR1614439 1

    work page 1998

  19. [19]

    D. A. Klain, An Euler relation for valuations on polytopes, Adv. Math. 147(1)(1999), 1-34. 1

    work page 1999

  20. [20]

    Kra´ skiewicz and P

    W. Kra´ skiewicz and P. Pragacz, Foncteurs de Schubert, C. R. Acad. Sci. Paris S´ er. I Math. 304(1987), no. 9, 209-211; MR0883476 1, 1

    work page 1987

  21. [21]

    Kra´ skiewicz and P

    W. Kra´ skiewicz and P. Pragacz, Schubert functors and Schubert polynomials, European J. Combin. 25 (8)(2004), 1327-1344. 1

    work page 2004

  22. [22]

    Lascoux and M

    A. Lascoux and M. Sch¨ utzenberger, Polynˆ omes de Schubert, C. R. Acad. Sci. Paris S´ er. I, Math. 294(13)(1982), 447-450. 2.1 17

    work page 1982

  23. [23]

    Lascoux and M

    A. Lascoux and M. Sch¨ utzenberger, Tableaux and noncommutative Schubert polynomials, Funct. Anal. Appl. 23 (3)(1989), 223–225. 2.1

    work page 1989

  24. [24]

    Lascoux and M

    A. Lascoux and M. Sch¨ utzenberger, Keys and Standard Bases, Institute for Mathematics and Its Applications, vol. 19, 1990, p. 125. 2.1

    work page 1990

  25. [25]

    Magyar, Schubert polynomials and Bott-Samelson varieties, Comment

    P. Magyar, Schubert polynomials and Bott-Samelson varieties, Comment. Math. Helv., 73 (4)(1998), 603-636. 1

    work page 1998

  26. [26]

    Manecke and R

    S. Manecke and R. Sanyal, Coprime Ehrhart theory and counting free segments, Int. Math. Res. Not. (IMRN) 2023, no. 9, 7319-7332; MR4584700 1

    work page 2023

  27. [27]

    M´ esz´ aros, L

    K. M´ esz´ aros, L. Setiabrata, and A. St. Dizier, On the support of Grothendieck polynomials, Ann. Comb. 29(2025), 541–562. 1, 5.2, 5.2, 5.5, 5.6, 5.2

    work page 2025

  28. [28]

    M´ esz´ aros, L

    K. M´ esz´ aros, L. Setiabrata, and A. St. Dizier. An orthodontia formula for Grothendieck polynomials. Trans. Amer. Math. Soc. 375(2022), 1281-1303 5.2

    work page 2022

  29. [29]

    Monical, N

    C. Monical, N. Tokcan, and A. Yong, Newton polytopes in algebraic combinatorics, Selecta Math. (N.S.) 25(2019), no. 5, Paper No. 66, 37 pp.; MR4021852 1, 2.1

    work page 2019

  30. [30]

    Nill and G

    B. Nill and G. M. Ziegler, Projecting lattice polytopes without interior lattice points, Math. Oper. Res. 36(2011), no. 3, 462-467; MR2832401 1

    work page 2011

  31. [31]

    Oda, Convex Polytopes and Algebraic Geometry, Springer, Berlin, 1998

    T. Oda, Convex Polytopes and Algebraic Geometry, Springer, Berlin, 1998. 1

    work page 1998

  32. [32]

    Schrijver,Combinatorial optimization

    A. Schrijver,Combinatorial optimization. Polyhedra and efficiency. Vol. A, Algorithms and Combi- natorics, 24, A, Springer, Berlin, 2003; MR1956924 2.2, 4.1, 4.2, 5.2

    work page 2003

  33. [33]

    H. E. Scarf, Integral polyhedra in three space, Math. Oper. Res. 10(1985), no. 3, 403-438; MR0798388 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610065, P.R. China Email address:doujinren@stu.scu.edu.cn, fan@scu.edu.cn, kw liu@stu.scu.edu.cn

    work page 1985