Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Vertex models for Canonical Grothendieck polynomials and their duals

T0 review · 4 major / 5 minor · reviewed 2026-08-27 · deepseek-v4-flash

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

Pith's one-line read Four solvable vertex models compute canonical Grothendieck polynomials and their duals, and their RLL relations imply two Cauchy identities.

desk verdict New vertex-model representation for canonical Grothendieck polynomials; clean construction, but inversion-relation proofs have gaps that leave one Cauchy identity conditional. read the letter →

arxiv 2009.13172 v1 pith:IUPZZQY7 submitted 2020-09-28 math.CO math-phmath.MP

Ajeeth Gunna , Paul Zinn-Justin This is my paper · ORCID
classification math.COmath-phmath.MP MSC 05E0582B23
keywords canonicalGrothendieckpolynomialsdualvertexmodelsintegrablelatticeRLLrelationsCauchyidentitiesinversionsymmetricfunctions
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 constructs solvable lattice models whose row-to-row transfer matrices compute the canonical Grothendieck polynomials $G^{(\alpha,\beta)}_\lambda$ and their Hall-dual polynomials $g^{(\alpha,\beta)}_\lambda$, in both row-multiplicity and column-multiplicity encodings of partitions. It proves that these transfer matrices satisfy RLL (Yang\textendash Baxter) relations, and from them derives an inversion relation between the row and column models together with two Cauchy identities: $\sum_\lambda G^{(-\alpha,-\beta)}_\lambda(x)\,g^{(\alpha,\beta)}_\lambda(y)=\prod_{i,j}(1-x_i y_j)^{-1}$ and $\sum_\lambda G^{(-\beta,-\alpha)}_{\lambda'}(x)\,g^{(\alpha,\beta)}_\lambda(y)=\prod_{i,j}(1+x_i y_j)$. If the paper is right, these identities are not isolated coefficient facts but consequences of the commutation of transfer matrices, and the known $\alpha=0,\beta=1$ dual Grothendieck Cauchy identity falls out by specialization. A reader should care because the lattice-model viewpoint packages the two-parameter families and their duality into a single integrable mechanism.

What carries the argument

The object that does the work is the transfer matrix $T(x)$ (or its row/column analogue) built from a local $L$-matrix and an $R$-matrix satisfying the RLL relation $R_{ij}(x,y)L_i(x)L_j(y)=L_j(y)L_i(x)R_{ij}(x,y)$. Each $R$-matrix has the column-sum property that the partition function of one vertex with a fixed right boundary is 1, which converts a local RLL move into a commutation relation $T(x)T(y)=T(y)T(x)$. The inversion relation is proved by a two-vertex cancellation: when the vertical boundary labels differ, the two allowed local configurations contribute weights that sum to zero, and when they agree the weight is exactly 1; this is the mechanism behind the duality between row and column encodings, i.e. between the involution $\omega$ and transposition of partitions.

What would settle it

Set $m=n=1$ in (58) and expand both sides as formal power series in $x$ and $y$; the coefficient of $x^k y^k$ on the right is 1 for every $k\ge 0$, so a single $k$ where the truncated left-hand sum over partitions gives a different coefficient would falsify the central identification. Concretely, one can compute the transfer-matrix value for $\lambda=(k)$ and its dual from the weights (4)/(19) and check the $k=1,2,3$ coefficients.

Watch

Extended reading notes

Core claim

The central claim is that four vertex models reproduce the two-parameter polynomial families on the nose: the fermionic row model with weights (4) and the bosonic column model with weights (28) both have transfer matrices whose matrix elements are $G^{(\alpha,\beta)}_\lambda(x_1,\dots,x_n)$, while the bosonic row model with weights (19) and the bosonic column model with weights (34) both produce $g^{(\alpha,\beta)}_\lambda(x_1,\dots,x_n)$. Theorems 1\textendash 4 prove this by showing that each transfer matrix obeys the same one-variable branching rule as the corresponding polynomial, so the two are equal after inducting on the number of variables. The RLL relations then imply the inversion relation $T(-x)\widetilde{T}\big(x/(1+(\alpha-\beta)x)\big)=1$ joining row and column transfer matrices, and the commutation relation $t(y)T^*(x)=\frac{1}{1-xy}T^*(x)t(y)$ between the dual row model and the canonical row model. From these, Theorems 6 and 7 derive the two Cauchy identities, and specializing parameters recovers the classical Grothendieck/dual-Grothendieck Cauchy identity and identities for generalised polynomials with inhomogeneities $z_i$.

Load-bearing premise

The load-bearing premise is that the branching formulas for $G^{(\alpha,\beta)}_\lambda$ and $g^{(\alpha,\beta)}_\lambda$ quoted from earlier work are correct and apply at every parameter value used here, since the paper does not re-derive them and every theorem identifying a transfer matrix with a polynomial feeds through those formulas.

Editorial extensions

If this is right

  • For all $\alpha,\beta$, the identity $\sum_\lambda G^{(-\alpha,-\beta)}_\lambda(x)\,g^{(\alpha,\beta)}_\lambda(y)=\prod_{i,j}(1-x_i y_j)^{-1}$ holds, and at $\alpha=0,\beta=1$ it reduces to the known finite-sum Cauchy identity for Grothendieck polynomials and their duals.
  • The second identity $\sum_\lambda G^{(-\beta,-\alpha)}_{\lambda'}(x)\,g^{(\alpha,\beta)}_\lambda(y)=\prod_{i,j}(1+x_i y_j)$ holds, giving a product with plus signs; its $\beta=0,\alpha=1$ specialization is a Cauchy identity for generalised weak Grothendieck and weak dual Grothendieck polynomials.
  • The inversion relation $T(-x)\widetilde{T}(x/(1+(\alpha-\beta)x))=1$ makes the symmetry $\omega(G^{(\alpha,\beta)}_\lambda)=G^{(\beta,\alpha)}_{\lambda'}$ visible at the level of transfer matrices, so the row and column models are not merely two encodings but are inverse objects.
  • With vertical inhomogeneities $z_i$, the generalised polynomials satisfy the same Cauchy kernels, and the specialization $x_i=z_i$ gives $G_\lambda(z_1,\dots,z_m;z_1,\dots,z_m)=1$, which yields a summation identity for dual Grothendieck polynomials over partitions of length at most $m$.

Reading between the lines

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

  • Beyond the paper: because the row model satisfies the difference property only for $\beta=0$ and the column model only for $\alpha=0$, the two models may be the two limits of a single inhomogeneous model with both parameters active; the paper does not construct such an interpolating model.
  • Beyond the paper: the column-sum eigenvector property of every $R$-matrix is a strong constraint; a natural testable extension is to perturb the Boltzmann weights while preserving this property and check whether the Cauchy kernel changes from $\prod(1-x_i y_j)^{-1}$ to another rational function.
  • Beyond the paper: the skew versions of the Cauchy identities mentioned in Section 6 suggest that the same RLL mechanism could prove skew-orthogonality relations for $G^{(\alpha,\beta)}$ and $g^{(\alpha,\beta)}$ with arbitrary skew shapes, which would connect to open problems in $K$-theoretic Schubert calculus.
  • Beyond the paper: the transfer matrices are presented as deformations of the free-fermionic vertex operators for Schur functions; if the inversion relation continues to hold after adding a spectral-parameter shift, one could obtain double or factorial analogues of the Cauchy identities, but this is not established here.
Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

  1. Claim #1: The central claim is that four vertex models reproduce the two-parameter polynomial families on the nose: the fermionic row model with weights (4) and the bosonic column model with weights (28) both have transfer matrices whose matrix elements are $G^{(\alpha,\beta)}_\lambda(x_1,\dots,x_n)$, while the bosonic row model with weights (19) and the bosonic column model with weights (34) both produce $

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper constructs exact solvable lattice models for canonical Grothendieck polynomials G^{(α,β)}_{λ} and their duals g^{(α,β)}_{λ}, in both row and column encodings of partitions. Theorems 1–4 identify the partition functions of the four vertex models with the corresponding polynomials by matching Yeliussizov's branching formulas. The paper then proves inversion relations between row and column transfer matrices (Props. 1–2), derives RLL commutation relations verified by case computations in the appendix, and uses them to prove two Cauchy identities (Thms. 6–7) as well as generalizations with inhomogeneous variables (Thms. 8 and Corollaries 1–3).

Significance. If the proofs are completed, the paper gives a unified integrable-lattice proof of the two Cauchy identities for canonical Grothendieck polynomials and their duals, recovering known identities as specializations. The explicit Boltzmann weights and the extensive case-by-case verification of the RLL relations in the appendix are a substantial strength, as is the use of inversion relations to interchange row and column models, which is conceptually novel. The main results depend on Yeliussizov's branching formulas, which are cited rather than re-derived; this is a legitimate external input but should be verified by the authors in the specific parameter ranges used. The paper is well organized and the examples are helpful, though several proofs, especially Proposition 1 and the boundary-sliding arguments, are not fully written out.

major comments (4)
  1. [Section 5, Proposition 1 (Eq. 53)] The induction step in the proof of Proposition 1 is not fully justified. The single-site cancellation is shown when the right horizontal edge is fixed to 0, but in a row of n sites the right edge of site 1 is the left edge of site 2, so it is not fixed. The text says one can 'move the free boundary condition across the physical line at site 1' and then apply induction, but the summation over the two possible horizontal labels at the cut is not written out. Since Eq. (53) is used to derive Eq. (54) and then Theorem 7, this gap directly affects one of the two headline Cauchy identities. Please provide an explicit argument for the boundary-sliding step, including the summation over the intermediate horizontal label.
  2. [Section 5, Proposition 2] The b = 0 case in the proof of Proposition 2 is handled by adding the weight of a configuration with a unique right boundary to the weight of a second configuration, and the text says 'we can get away with it.' This indicates that the free-boundary-sliding argument is not automatic for b = 0. The proof should explicitly show that the combined weight factors as the transfer matrix of size n, including the case where the right boundary label is not free. As written, the argument is a sketch rather than a proof, and the same issue affects the derivation of the inversion relation used in Theorem 8.
  3. [Section 6, Theorems 6 and 7] The proof of Theorem 7 relies on the inversion relation eq. (54) combined with the commutation relation from Proposition 3. While the RLL relation is checked by case analysis in Appendix A.4, the derivation of eq. (54) from Proposition 1 is asserted without proof. Because the inversion relation is the only bridge between the row and column models in this argument, the proof of Theorem 7 is conditional on completing the proof of Proposition 1. Please either complete the inversion-relation proof or state Theorem 7 as conditional on it.
  4. [Section 2.2, Theorems 1 and 3] The identification of the vertex model partition functions with G^{(α,β)}_{λ} depends on the branching formula (15)–(16) cited from Yeliussizov [Yel17, Prop. 8.8]. The paper does not re-derive this formula or verify that its hypotheses (e.g., conditions on α, β for the polynomials to be well-defined) are satisfied in the parameter ranges used later, especially in the generalized cases of Section 4. This is not a circularity, but it is a load-bearing external input; please state explicitly which parameter ranges are assumed and confirm that the cited branching formulas apply verbatim.
minor comments (5)
  1. [Section 2.3, Eq. (19)] In the second case of the Boltzmann weights, the condition is written as '0 < a ≤ d'; the first case already covers a > d, so the second case should be stated as '0 < a ≤ d' rather than 'a ≤ d' to avoid ambiguity with a = 0, which is treated separately.
  2. [Section 3.4, Eq. (48)] There is a typo in the sentence introducing the dual transfer matrix: 'ths weight' should be 'the weight'.
  3. [Section 6, Proof of Theorem 7] The displayed product in the penultimate equation uses the bounds '1≤n, 1≤m' instead of '1≤i≤n, 1≤j≤m' and omits the summation indices; please correct these bounds.
  4. [Section 4.2, Eq. (51)] The paper defines R(y/x) but does not explicitly state that this is the same R-matrix as in Eq. (9) after substituting β = 0 and rescaling the spectral parameter. A short explanation would improve readability.
  5. [Appendix A.1.5] The unitary relation for the ~R matrix is presented with a diagram but the text says 'the ~R stisfies' — presumably 'satisfies'. Please correct the typo and state the relation explicitly as an equation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the vertex-model partition functions are matched to Yeliussizov's branching formulas, and the Cauchy identities are derived from explicit RLL and inversion relations.

full rationale

The derivation chain is self-contained with respect to the target identities. The four transfer matrices are defined by explicit Boltzmann weights (eqs. 4, 19, 28, 34). Theorems 1–4 identify their partition functions with the canonical Grothendieck polynomials and duals by matching the branching formulas quoted from Yeliussizov ([Yel17, Prop. 8.8, Thm. 8.6], eqs. 15–16 and 25); these cited formulas serve as external background definitions, not as circular support for the paper's own claims, and they do not presuppose the Cauchy identities. The inversion relations (Prop. 1, eq. 53, and Prop. 2) are proved from the local weights by explicit cancellation in the size-one case followed by an induction on the number of sites; the RLL relations used to prove the commuting transfer-matrix properties are verified entrywise in Appendix A. The Cauchy identities (Thm. 6, eq. 58, and Thm. 7, eq. 60) are then obtained by repeatedly applying the commutation relations derived from RLL, together with the inversion relations. The final products, 1/(1-x_i y_j) and (1+x_i y_j), are never assumed as inputs; no parameter is fitted to any subset of the identities. Self-citations in the paper (e.g., [GZJ], [ZJ09], [WZJ19]) are contextual or motivational and are not load-bearing: [GZJ] is explicitly described as work in progress and is not used in any proof. The only caveats worth noting are rigor concerns, not circularity: the induction step in Proposition 1 involving moving the free boundary condition is sketched rather than fully written out, and Proposition 2 contains the phrase 'we can get away with it' when handling the b=0 case. These are potential gaps in the proof of the inversion relations and would affect correctness, not the independence of the derivation; they do not make the conclusion equivalent to its inputs. Overall, no circular step was found.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical entities or fitted parameters; the R-matrices and L-matrices are explicit functions of the existing parameters alpha, beta and spectral variables. The only external inputs are the combinatorial definitions and branching formulas of canonical Grothendieck polynomials from Yeliussizov.

assumptions (3)
  • domain assumption Branching formulas for G^{(alpha,beta)}_lambda (eqs. 15-16) and g^{(alpha,beta)}_lambda (eq. 25) from Yeliussizov [Yel17].
    Used as the definition and benchmark in Theorems 1-4 to prove the vertex model representation; accepted from prior literature without proof.
  • domain assumption Involution relations omega(G^{(alpha,beta)}_lambda) = G^{(beta,alpha)}_{lambda'} and omega(g^{(alpha,beta)}_lambda) = g^{(beta,alpha)}_{lambda'} (Section 1).
    Used to identify j-polynomials and in the lambda' formulation of the second Cauchy identity (Thm. 7); cited to Yeliussizov.
  • domain assumption The infinite transfer matrix limits in eqs. (12)-(14) are entry-wise well-defined.
    The paper argues this from the uniqueness of finite configurations with fixed right boundary; listed as an assumption since no fully rigorous convergence proof is given.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Vertex models for Canonical Grothendieck polynomials and their duals." pith.science (2026). https://pith.science/paper/IUPZZQY7

@misc{pith2026200913172,
  author       = {Pith},
  title        = {Pith review of: Vertex models for Canonical Grothendieck polynomials and their duals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IUPZZQY7}},
  note         = {Machine review of arXiv:2009.13172}
}
read the original abstract

We study solvable lattice models associated to canonical Grothendieck polynomials and their duals. We derive inversion relations and Cauchy identities.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Lung Disease Detection with Vision Transformers: A Comparative Study of Machine Learning Methods

    eess.IV 2024-11 reject novelty 3.0 of 10

    A Vision Transformer on chest X-rays is claimed to reach 97.83% accuracy, but the reported validation accuracy and missing baselines do not support this claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

  1. [1]

    Valentin Buciumas and Travis Scrimshaw, Double G rothendieck polynomials and colored lattice models , 2020, http://arxiv.org/abs/2007.04533 arXiv:2007.04533

    work page Pith review arXiv 2020

  2. [2]

    Yang-Baxter equation, symmetric functions and Grothendieck polynomials

    S. Fomin and A. Kirillov, Y ang-- B axter equation, symmetric functions and G rothendieck polynomials , 1993, http://arxiv.org/abs/hep-th/9306005 arXiv:hep-th/9306005

    work page Pith review arXiv 1993

  3. [3]

    Ajeeth Gunna and Paul Zinn-Justin, work in progress

    work page

  4. [4]

    Interacting dimers on the honeycomb lattice: An exact solution of the five-vertex model

    H.Y. Huang, F.Y. Wu, H. Kunz, and D. Kim, Interacting dimers on the honeycomb lattice: an exact solution of the five-vertex model, Physica A 228 (1996), 1--32, http://arxiv.org/abs/cond-mat/9510161 arXiv:cond-mat/9510161

    work page Pith review arXiv 1996

  5. [5]

    Schubert puzzles and integrability I: invariant trilinear forms

    A. Knutson and P. Zinn-Justin, Schubert puzzles and integrability I : invariant trilinear forms , 2017, http://arxiv.org/abs/1706.10019 arXiv:1706.10019

    work page Pith review arXiv 2017

  6. [6]

    Japan Acad

    Alain Lascoux and Hiroshi Naruse, Finite sum C auchy identity for dual G rothendieck polynomials , Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 7, 87--91, http://dx.doi.org/10.3792/pjaa.90.87 doi:10.3792/pjaa.90.87

    work page doi:10.3792/pjaa.90.87 2014

  7. [7]

    Combinatorial Hopf algebras and K-homology of Grassmanians

    T. Lam and P. Pylyavskyy , Combinatorial H opf A lgebras and K - H omology of G rassmanians , International Mathematics Research Notices 2007 (2007), no. 9, rnm125, http://arxiv.org/abs/0705.2189 arXiv:0705.2189

    work page Pith review arXiv 2007

  8. [8]

    118--144

    Alain Lascoux and Marcel-Paul Sch \"u tzenberger, Symmetry and flag manifolds, Invariant Theory (Berlin, Heidelberg) (Francesco Gherardelli, ed.), Springer Berlin Heidelberg, 1983, pp. 118--144

    work page 1983

Show all 17 references
  1. [9]

    McNamara, Factorial G rothendieck polynomials , Electron

    Peter J. McNamara, Factorial G rothendieck polynomials , Electron. J. Combin. 13 (2006), no. 1, Research Paper 71, 40, http://arxiv.org/abs/math/0508192 arXiv:math/0508192 , http://dx.doi.org/10.37236/1097 doi:10.37236/1097 . 2240776

    2006 arXiv

  2. [10]

    Kohei Motegi and Kazumitsu Sakai, K-theoretic boson-fermion correspondence and melting crystals, Journal of Physics A: Mathematical and Theoretical 47 (2013), http://arxiv.org/abs/1311.6076 arXiv:1311.6076 , http://dx.doi.org/10.1088/1751-8113/47/44/445202 doi:10.1088/1751-811...

    2013 arXiv

  3. [11]

    , Vertex models, TASEP and G rothendieck polynomials , Journal of Physics A: Mathematical and Theoretical 46 (2013), http://arxiv.org/abs/1305.3030 arXiv:1305.3030 , http://dx.doi.org/10.1088/1751-8113/46/35/355201 doi:10.1088/1751-8113/46/35/355201

    2013 arXiv

  4. [12]

    Wheeler and P

    M. Wheeler and P. Zinn-Justin, L ittlewood-- R ichardson coefficients for G rothendieck polynomials from integrability , J. Reine Angew. Math. 757 (2019), 159--195, http://arxiv.org/abs/1607.02396 arXiv:1607.02396 , http://dx.doi.org/10.1515/crelle-2017-0033 doi:10.1515/crelle...

    2019 arXiv

  5. [13]

    1, 295--344, http://arxiv.org/abs/1601.01581 arXiv:1601.01581 , http://dx.doi.org/10.1007/s10801-016-0708-4 doi:10.1007/s10801-016-0708-4

    Damir Yeliussizov, Duality and deformations of stable G rothendieck polynomials , Journal of Algebraic Combinatorics 45 (2017), no. 1, 295--344, http://arxiv.org/abs/1601.01581 arXiv:1601.01581 , http://dx.doi.org/10.1007/s10801-016-0708-4 doi:10.1007/s10801-016-0708-4

    2017 arXiv

  6. [14]

    , E numeration of plane partitions by descents , 2019, http://arxiv.org/abs/1911.03259 arXiv:1911.03259

    2019 arXiv

  7. [15]

    , Symmetric G rothendieck polynomials, skew C auchy identities, and dual filtered Y oung graphs , J. Comb. Theory, Ser. A 161 (2019), 453--485, http://arxiv.org/abs/1711.09544 arXiv:1711.09544

    2019 arXiv

  8. [16]

    P. Zinn-Justin, Six-vertex, loop and tiling models: integrability and combinatorics, Lambert Academic Publishing, 2009, Habilitation thesis, http://www.lpthe.jussieu.fr/ pzinn/publi/hdr.pdf

    2009

  9. [17]

    , Schur functions and L ittlewood-- R ichardson rule from exactly solvable tiling models , 2012, Chern--Simons Research Lectures, http://www.lpthe.jussieu.fr/ pzinn/semi/berkeley.pdf

    2012

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.