Pith. sign in

REVIEW 3 major objections 4 minor 26 references

Tableau formula for vexillary double Edelman--Greene coefficients

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Vexillary double Edelman–Greene coefficients admit a manifestly positive tableau expansion.

desk verdict Genuinely new refined Graham positivity for vexillary double Edelman–Greene coefficients; proof is plausible but requires verifying the imported formal-β flagged equality. read the letter →

arxiv 2412.20615 v1 pith:4YSOUWDR submitted 2024-12-29 math.CO

Adam Gregory , Zachary Hamaker , Tianyi Yu This is my paper · ORCID
classification math.CO MSC 05E0505E1014M15
keywords vexillarypermutationsdoubleEdelman-Greenecoefficientsbeta-Grahampositivityflaggedset-valuedtableauxbackstableGrothendieckpolynomialsbeta-StanleysymmetricfunctionsK-theorySchubertcalculus
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

For a vexillary permutation $w\in S_{\mathbb{Z}}$, the paper proves that the double $\beta$-Edelman--Greene coefficient $j^w_\mu(\beta;y)$, multiplied by $\beta^{\ell(w)-|\mu|}$, is a sum of monomials indexed by flagged set-valued tableaux. Every monomial is a product of factors $\beta(y_i\ominus y_j)$ with positive coefficient, so the expansion is manifestly $\beta$-Graham positive. The formula is finer than previous positivity statements: in each monomial, a factor with $0

What carries the argument

The central object is a flagged set-valued semistandard tableau (an $S$-tableau) with monomial weight $\beta(x_i\ominus y_j)$ per entry, and the load-bearing mechanism is the split of such a tableau into a non-positive part $T^-$ and a positive part $T^+$. Lemma 3.2 makes this split a bijection between $\operatorname{SetSSYT}^{\varphi}(\lambda)$ and a disjoint union of products $\operatorname{SetSSYT}^{\varphi^-}(\nu)\times \operatorname{SetSSYT}^{\varphi^+}(\lambda/\mu)$ with disconnected skew piece, and Corollary 3.3 converts it into the convolution formula $j^{\lambda,\varphi}_\rho=\sum_\nu j^{\nu,\varphi^-}_\rho\, j^{\lambda,\varphi^+}_\nu$. Positivity of each factor is established by separating cells above and below the diagonal, by a custom permutation $\pi$ of the $x$-variables (Lemma 3.7), and by an extension $\tilde\omega$ of the $\omega$ involution that swaps non-positive and non-negative flags.

What would settle it

For the vexillary permutation w=345162, with λ(w)=(2,2,2,1) and φ(w)=(3,3,3,5), compute j^w_(2,2)(β;y) both by direct expansion of the backstable double β-Grothendieck polynomial and by the paper's tableau formula; a monomial containing both a Type 1 and a Type 2 factor, or any Type 3 factor more than twice, would disprove Theorem 1.1. The same comparison at β=0 against the double Schur expansion of the backstable Schubert polynomial would test the specialized corollary.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1, sharpened as Theorem 4.7: for every vexillary $w\in S_{\mathbb{Z}}$ and every partition $\mu$, $\beta^{\ell(w)-|\mu|} j^w_\mu(\beta;y)$ is a sum of monomials $\prod \beta(y_i\ominus y_j)$, with each monomial containing at most one of the two one-sided factor types (Type 1: $0<i<j$; Type 2: $i<j\le 0$) and at most two copies of any Type 3 factor ($j\le 0<i$). The proof is combinatorial: it reduces the backstable double $\beta$-Grothendieck polynomial of a vexillary permutation to the flagged double stable $\beta$-Grothendieck function $G^{\varphi(w)}_{\lambda(w)}(\beta;x;y)$, decomposes the set-valued tableaux into positive and non-positive parts, and analyzes the resulting skew shapes above and below the diagonal. The $\beta=0$ specialization yields the first manifestly Graham-positive tableau rule for the double Edelman--Greene coefficients of vexillary permutations.

Load-bearing premise

The whole argument rests on the imported equality (Proposition 2.5, from [12, Theorem 5.8]) saying that for a vexillary permutation the backstable double β-Grothendieck polynomial equals the flagged double β-Grothendieck function; if that equality fails in the backstable infinite setting, the tableau formula would not compute the intended coefficients.

Editorial extensions

If this is right

  • Every vexillary double $\beta$-Edelman--Greene coefficient is a positive integer combination of $\beta$-Graham monomials, so its expansion has no cancellation.
  • The refined type restriction is new: no monomial contains both Type 1 and Type 2 factors, Type 1 or Type 2 factors appear at most once, and Type 3 factors appear at most twice.
  • At $\beta=0$, the formula gives the first manifestly Graham-positive tableau rule for double Edelman--Greene coefficients, including the vexillary double Schur coefficients.
  • The formula reduces nonvanishing of $j^w_\mu$ to a polynomial-time check when $\mu\subseteq\lambda(w)$ and $\lambda(w)/\mu$ has no diagonal cells, as the paper notes in Section 5.3.
  • The convolution formula factors any vexillary coefficient as $j^{\lambda,\varphi}_\rho=j^{\nu,\varphi^-}_\rho\, j^{\lambda,\varphi^+}_\nu$ for the unique $\nu$ selected by Corollary 4.3.

Reading between the lines

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

  • As an extension beyond the paper, if the type-restricted positivity is preserved under the transition equations the authors sketch, the theorem would propagate beyond vexillary permutations; the introduction's non-vexillary example shows the full Type 1/Type 2 separation cannot hold universally.
  • The constructive permutation $\pi$ in Lemma 3.7 can be implemented as an algorithm that outputs the monomials directly, so one could compare its output term-by-term with the nonconstructive geometric positivity witness.
  • The same positive/non-positive tableau split may yield a tableau rule for equivariant $K$-theory Grassmannian structure coefficients, since the paper's final remarks connect double $\beta$-Edelman--Greene coefficients to equivariant $K$-homology.
  • A direct empirical check is whether the 'at most twice' bound on Type 3 factors is ever attained; the paper does not exhibit an example of a monomial with a repeated Type 3 factor, so finding one would show the bound is sharp.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper claims a tableau formula for vexillary double β–Edelman–Greene coefficients j^w_µ(β;y), refining Anderson's β-Graham positivity theorem. The authors reduce the problem via Proposition 2.5 to flagged double stable β-Grothendieck functions, decompose flagged set-valued tableaux according to positive and non-positive entries, apply a Bender-Knuth symmetry argument to separate above- and below-diagonal parts, and then combine the two pieces using the ω involution. The main results, Theorem 1.1 and Theorem 4.7, assert that after multiplying by β^{ℓ(w)-|µ|}, each coefficient is a sum of β-Graham positive monomials, with a finer control on the types of terms that can occur: Type 3 terms appear at most twice per monomial, Type 1 or Type 2 at most once, and no monomial mixes Type 1 with Type 2.

Significance. If correct, this is a substantial combinatorial advance: it provides the first manifestly Graham-positive tableau description of double Edelman-Greene coefficients in the vexillary case, including the β=0 specialization, and it refines a geometric positivity result of Anderson. The proof is largely self-contained once the imported equality of Proposition 2.5 is granted, and it combines several interesting ingredients: a sign-based decomposition of set-valued tableaux, a Bender-Knuth symmetry for flagged skew functions, and an ω-involution step. The manuscript also includes useful appendix material establishing that backstable double β-Grothendieck polynomials expand into double stable β-Grothendieck functions. The main caveats are that Proposition 2.5 depends crucially on a cited theorem whose exact scope is not made explicit, and that two key lemmas (3.10 and 3.14) are compressed and contain apparent indexing errors. These issues are local and likely fixable, but they are load-bearing for the main theorem.

major comments (3)
  1. [§2.3, Proposition 2.5] The entire reduction to tableaux hinges on the assertion that for vexillary w in S_Z, the backstable double β-Grothendieck polynomial equals the flagged double stable β-Grothendieck function. The proof says this is 'a straightforward consequence of [12, Thm 5.8]' and refers to a 'double Schubert polynomial equality' G_w(β;x_+;y_+) = G^{φ(w)}_{λ(w)}(β;x_+;y_+). Since Theorem 1.1 concerns formal β and the backstable infinite setting, the cited theorem must cover the β-Grothendieck polynomial defined in (8), not only the β=0 double Schubert polynomial or the β=-1 K-theory specialization, and it must use the same flag and shape conventions. Please state precisely what [12, Thm 5.8] proves, verify that it applies to the β-deformed functions in this paper, and spell out the γ^{-p} shift argument for S_Z. If [12] only treats β=0, then (9) is unsupported and the tableau model in Section 3 does not compute the double β-Edelman-Greene coefficients of Theorem 1.1.
  2. [§3.2, Lemma 3.14] Step 2 of Lemma 3.14 contains an indexing error that affects the proof of Lemma 3.7. Since χ(i) and χ(i+1) differ at row i+1, the violating eigenvalue bound should be ψ_{i+1} < k ≤ φ_{i+1}, not ψ_{i+1} < k ≤ φ_i; the text currently writes φ_i. Likewise, when ψ_{i+1} ≠ φ_{i+1}, the displayed identity should be ψ_{i+1} = i+1 - λ_{i+1} (from the definition ψ_i = min(i-λ_i, φ_i)), not ψ_{i+1} = i - λ_{i+1}. The subsequent statement that π_{i+1} sends j to j - ψ_{i+1} = j - (i+1) + λ_{i+1} for ψ_{i+1} < j ≤ φ_i also needs the upper bound corrected to φ_{i+1}. As written, the proof does not establish the equality (16) for all values in the relevant range. Please correct and reprove this step.
  3. [§3.2, Lemma 3.10] The Bender-Knuth argument proving symmetry in xi,...,xj is too compressed to be fully verifiable. The proof asserts a partition of tableaux into classes A_k with frozen entries and row segments, but it does not define the involution on free entries explicitly or check that the flag constraints and semistandard conditions are preserved when i and i+1 are swapped. Since Lemma 3.10 is used in Lemma 3.14 Step 1 to justify the symmetry that allows replacing π_i by π_{i+1}, this is a load-bearing step. Please expand the proof, ideally by giving the precise involution and verifying that it respects SetSSYT^φ_+(λ/µ).
minor comments (4)
  1. [Abstract] The abstract contains a small grammatical error: 'The goal of this paper to understand' should read 'The goal of this paper is to understand'.
  2. [§2.3, Proposition 2.5 proof] The phrase 'double Schubert polynomial equality' is inconsistent with the β-deformed setting; this should be 'double β-Grothendieck polynomial equality' or the scope should be clarified.
  3. [§5.1] The claim that w_{λ,φ} is 'the unique vexillary permutation with shape λ and flag φ' should be justified or given a reference, since Proposition 2.3 only guarantees existence up to flag equivalence.
  4. [§5.5] The sentence 'It would be an accomplishment to recover equivariant Grassmannian cohomology [13] or K-theory structure coefficients [21] in this way' is a bit vague; consider being more specific about what a recovery would mean, since those coefficients already have combinatorial rules.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the tableau positivity proof rests on an imported external identification and internal combinatorial arguments, not on its own conclusion.

full rationale

The derivation chain is self-contained against independently established inputs. The central reduction is Proposition 2.5, an imported equality from [12, Thm 5.8] relating a vexillary backstable double beta-Grothendieck polynomial to a flagged double beta-Grothendieck function; this is a nontrivial external theorem, not a restatement of Theorem 1.1 or of Anderson's positivity result. Equation (9), identifying j^w_mu with j^{lambda(w),phi(w)}_mu, is a consequence of that imported equality and of the coefficient definitions, not a definitional identity. The paper's own contributions, including the set-valued tableau decomposition (Lemma 3.2), the coefficient convolution identity (Proposition 3.5), and the Type 1/2/3 positivity analysis (Lemmas 3.6, 3.8, 4.5, 4.6), are proved by explicit bijections, polynomial identities, and vanishing arguments that never assume the positivity being proved. No parameter is fitted to target data, no prediction is renamed input, and the proof does not depend on a load-bearing self-citation chain; the cited works [1], [12], [14], [17], and [24] are external prior results. Potential questions about the exact scope of the imported theorem would be correctness concerns, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities. The proof imports standard structural results from the cited literature; the new constructions (psi flag and permutation pi) are defined and proved within the paper.

assumptions (4)
  • standard math Backstable double beta-Grothendieck polynomials expand into double stable Grothendieck functions (expansion (1) and Proposition 2.7).
    Imported from Lam-Lee-Shimozono [14] and proved in Appendix A; used to define the coefficients and the reduction to flagged functions.
  • domain assumption For w vexillary, the backstable polynomial equals the flagged double beta-Grothendieck function G^{phi(w)}_{lambda(w)} (Proposition 2.5).
    From [12, Thm 5.8] extended to S_Z; this is the central reduction of the proof.
  • domain assumption Compatibility condition (6) and the bijection with vexillary permutations (Proposition 2.3).
    From Macdonald [17] and Wachs [24]; guarantees the flagged data arise from a permutation.
  • domain assumption The omega involution lemma (Lemma 2.8) for backstable functions.
    From [14, Prop 5.23]; used to transfer positivity from non-negative to non-positive flags.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Tableau formula for vexillary double Edelman--Greene coefficients." pith.science (2026). https://pith.science/paper/4YSOUWDR

@misc{pith2026241220615,
  author       = {Pith},
  title        = {Pith review of: Tableau formula for vexillary double Edelman--Greene coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4YSOUWDR}},
  note         = {Machine review of arXiv:2412.20615}
}
abstract

Lam, Lee and Shimozono recently introduced backstable double Grothendieck polynomials to represent $K$-theory classes of the infinite flag variety. They used them to define double $\beta$-Stanley symmetric functions, which expand into double stable Grothendieck functions with polynomial coefficients called double $\beta$-Edelman--Greene coefficients. Anderson proved these coefficients are $\beta$-Graham positive. For vexillary permutations, this is equivalent to a statement for skew flagged double $\beta$-Grothendieck functions. Working in this setting, we give a tableau formula for vexillary double $\beta$-Edelman--Greene coefficients that is manifestly $\beta$-Graham positive. Our formula demonstrates a finer notion of positivity than was previously known.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 25 canonical work pages

  1. [12]

    Gr¨ obner geom etry of vertex decompositions and of flagged tableaux

    Allen Knutson, Ezra Miller, and Alexander Yong. “Gr¨ obner geom etry of vertex decompositions and of flagged tableaux”. In: J. Reine Angew. Math. 630 (2009), pp. 1–31. 2More commonly referred to as the stable Grothendieck polyno mial of w. REFERENCES 15

    work page 2009

  2. [1]

    Strong equivariant positivity for homogeneous varieties a nd back-stable coproduct coefficients

    David Anderson. Strong equivariant positivity for homogeneous varieties a nd back-stable coproduct coefficients . 2023. arXiv: 2302.12765 [math.AG]

    arXiv 2023

  3. [2]

    Chern class formulas for classic al-type degeneracy loci

    David Anderson and William Fulton. “Chern class formulas for classic al-type degeneracy loci”. In: Compositio Mathematica 154.8 (2018), pp. 1746–1774

    work page 2018

  4. [3]

    Degeneracy Loci, Pfaffians, and Vexillary Signed Permutations in Types B, C, and D

    David Anderson and William Fulton. “Degeneracy loci, Pfaffians, and vexillary signed permutations in types B, C, and D”. In: arXiv preprint arXiv:1210.2066 (2012)

    work page Pith review arXiv 2012

  5. [4]

    Permutation patterns, Stan ley symmetric functions, and gener- alized Specht modules

    Sara Billey and Brendan Pawlowski. “Permutation patterns, Stan ley symmetric functions, and gener- alized Specht modules”. In: Journal of Combinatorial Theory, Series A 127 (2014), pp. 85–120

    work page 2014

  6. [5]

    Stable Grothendieck polynomials an d K-theoretic factor sequences

    Anders Skovsted Buch et al. “Stable Grothendieck polynomials an d K-theoretic factor sequences”. In: Mathematische annalen 340 (2008), pp. 359–382

    work page 2008

  7. [6]

    A Littlewood-Richardson rule for the K-theory of Gra ssmannians

    AS Buch. “A Littlewood-Richardson rule for the K-theory of Gra ssmannians”. In: Acta Mathematica 189.1 (2002), pp. 37–78

    work page 2002

  8. [7]

    Balanced tableaux

    Paul Edelman and Curtis Greene. “Balanced tableaux”. In: Adv. in Math. 63.1 (1987), pp. 42–99

    work page 1987

Show all 26 references
  1. [8]

    Positivity in equivariant Schubert calculus

    William Graham. “Positivity in equivariant Schubert calculus”. In: Duke Math. J. 109.3 (2001), pp. 599–614

    2001

  2. [9]

    Schubert products for permutations with sepa rated descents

    Daoji Huang. “Schubert products for permutations with sepa rated descents”. In: International Math- ematics Research Notices 2023.20 (2023), pp. 17461–17493

    2023

  3. [10]

    Do uble Schubert polynomials for the clas- sical groups

    Takeshi Ikeda, Leonardo C Mihalcea, and Hiroshi Naruse. “Do uble Schubert polynomials for the clas- sical groups”. In: Advances in Mathematics 226.1 (2011), pp. 840–886

    2011

  4. [11]

    Construction of double Gr othendieck polynomials of classical types using idCoxeter algebras

    Anatol N Kirillov and Hiroshi Naruse. “Construction of double Gr othendieck polynomials of classical types using idCoxeter algebras”. In: Tokyo Journal of Mathematics 39.3 (2017), pp. 695–728

    2017

  5. [13]

    Puzzles and (equivariant) coh omology of Grassmannians

    Allen Knutson and Terence Tao. “Puzzles and (equivariant) coh omology of Grassmannians”. In: Duke Math. J. 120.3 (2003), pp. 221–260

    2003

  6. [14]

    Back stable K -theory Schubert calculus

    Thomas Lam, Seung Jin Lee, and Mark Shimozono. “Back stable K -theory Schubert calculus”. In: International Mathematics Research Notices 2023.24 (2023), pp. 21381–21466

    2023

  7. [15]

    Back stable S chubert calculus

    Thomas Lam, Seung Jin Lee, and Mark Shimozono. “Back stable S chubert calculus”. In: Compos. Math. 157.5 (2021), pp. 883–962

    2021

  8. [16]

    Transition on Grothendieck polynomials

    Alain Lascoux. “Transition on Grothendieck polynomials”. In: Physics and combinatorics . World Sci- entific, 2001, pp. 164–179

    2001

  9. [17]

    Notes on Schubert polynomials

    Ian G Macdonald. Notes on Schubert polynomials . LACIM, 1991

    1991

  10. [18]

    Principal specializations o f Schubert polynomials in classical types

    Eric Marberg and Brendan Pawlowski. “Principal specializations o f Schubert polynomials in classical types”. In: Algebraic Combinatorics 4.2 (2021), pp. 273–287

    2021

  11. [19]

    A Tableau Formula for Vexillary Schubert Po lynomials in Type C

    Tomoo Matsumura. “A Tableau Formula for Vexillary Schubert Po lynomials in Type C”. In: The Electronic Journal of Combinatorics (2023), P1–28

    2023

  12. [20]

    Comultiplication rules for the double Schur fun ctions and Cauchy identities

    Alexander Molev. “Comultiplication rules for the double Schur fun ctions and Cauchy identities”. In: The Electronic Journal of Combinatorics 16.R13 (1 2009), 44pp

    2009

  13. [21]

    Equivariant-theory of Gr assmannians

    Oliver Pechenik and Alexander Yong. “Equivariant-theory of Gr assmannians”. In: Forum of Mathe- matics, Pi 5 (2017), e3

    2017

  14. [22]

    Bruce E. Sagan. The symmetric group . Second. Vol. 203. Graduate Texts in Mathematics. Rep- resentations, combinatorial algorithms, and symmetric functions . Springer-Verlag, New York, 2001, pp. xvi+238. isbn: 0-387-95067-2

    2001

  15. [23]

    On the number of reduced decompositions of elements of Coxeter groups

    Richard P. Stanley. “On the number of reduced decompositions of elements of Coxeter groups”. In: European J. Combin. 5.4 (1984), pp. 359–372

    1984

  16. [24]

    Flagged Schur functions, Schubert polynomia ls, and symmetrizing operators

    Michelle L Wachs. “Flagged Schur functions, Schubert polynomia ls, and symmetrizing operators”. In: Journal of Combinatorial Theory, Series A 40.2 (1985), pp. 276–289

    1985

  17. [25]

    Bumpless pipe dreams and alternating sign matr ices

    Anna Weigandt. “Bumpless pipe dreams and alternating sign matr ices”. In: J. Combin. Theory Ser. A 182 (2021), Paper No. 105470, 52pp

    2021

  18. [26]

    Littlewood–Richardson coefficients for Grothendieck poly- nomials from integrability

    Michael Wheeler and Paul Zinn-Justin. “Littlewood–Richardson coefficients for Grothendieck poly- nomials from integrability”. In: Journal f¨ ur die reine und angewandte Mathematik (Crelles J ournal) 2019.757 (2019), pp. 159–195

    2019

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.