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Integrability approach to Feher-Nemethi-Rimanyi-Guo-Sun type identities for factorial Grothendieck polynomials

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Pith's one-line read By re-expressing five-vertex-model wavefunctions in two ways, this paper proves a new identity for factorial Grothendieck polynomials of rectangular shape and re-proves the Guo-Sun identity, with a duality formula and a q-deformation…

desk verdict A useful integrability-based proof of an existing identity plus a genuinely new rectangular identity, held back by one load-bearing commutation relation that is only sketched. read the letter →

arxiv 1909.02278 v2 pith:EMNJXPDN submitted 2019-09-05 math.CO math-phmath.MP

Kohei Motegi This is my paper · ORCID
classification math.COmath-phmath.MP MSC 05E0505E1081R1282B23
keywords factorialGrothendieckpolynomialsfive-vertexmodelquantuminversescatteringmethodGuo-SunidentityrectangularYoungdiagramsq-deformationYang-Baxterequationwavefunctions
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 establishes a new identity for factorial Grothendieck polynomials attached to rectangular Young diagrams, and gives an integrability-based proof of the Guo-Sun identity in the $\beta=-1$ case. Both results come from a single move: write the same five-vertex-model wavefunction in two ways, once through the known wavefunction-polynomial correspondence and once by commuting transfer-matrix operators with the quantum inverse scattering method, then equate the two expressions. If the argument is correct, the Guo-Sun identity and the new rectangular identity hold in full generality, and a duality formula and a $q$-deformation follow as well.

What carries the argument

The load-bearing object is the five-vertex model, obtained as the $q=0$ limit of the $U_q(\widehat{\mathfrak{sl}}_2)$ six-vertex $R$-matrix, together with the correspondence that equates its wavefunctions with (prefactors times) factorial Grothendieck polynomials. The argument runs on the quantum inverse scattering method: the relevant wavefunction is expressed once through this correspondence and once by isolating a forced configuration in part of the lattice and commuting the remaining $B$- and $D$-operators (or, in the rectangular case, the $A$- and $C$-operators) using the commutation relations implied by the Yang-Baxter equation. The compact form of those commutation relations, which the paper adapts from a prior analysis of an integrable phase model, is what turns the operator reordering into the polynomial sums of the identities.

What would settle it

For small values such as $m=2$, $k=1$, $n=2$, apply both sides of the commutation relation (4.10) to the reference state and compare the coefficients of the $C$ and $A$ operator products; equivalently, evaluate the rectangular identity (4.1) at generic complex numbers for both sides and check equality.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that for $\beta=-1$ the factorial Grothendieck polynomial of the rectangular partition $\mu=((m-k)^{n-k},0^k)$ satisfies the explicit identity (4.1), a sum over $k$-subsets $S$ of $\{1,\dots,m\}$ involving products of $(1-\alpha_i)^{m-k}$ and $z_i\oplus\alpha_j$ divided by Vandermonde-type differences $\alpha_j-\alpha_i$; and that the Guo-Sun identity (1.3) is a consequence of the same method. The proof identifies the polynomial (up to a prefactor) with a wavefunction of the five-vertex model, decomposes the wavefunction graphically into a frozen part and a remaining operator product, and uses Yang-Baxter commutation relations to reorder the operators. Equating the two evaluations produces the identities.

Load-bearing premise

The new rectangular identity rests on a compact commutation relation for the transfer-matrix operators that the paper takes from an earlier argument and only sketches; if that relation's coefficients or operator order are wrong, the identity does not follow.

Editorial extensions

If this is right

  • The Guo-Sun identity for $\beta=-1$ receives a new proof from quantum integrability, independent of the original derivation.
  • The new rectangular identity (Theorem 4.1) holds for all admissible $m,n,k$ with $0\le k\le m$.
  • Combining the two identities yields a duality formula (Theorem 4.2) that interchanges the roles of the spectral variables $z_i$ and the factorial parameters $\alpha_j$.
  • Running the same computation on the six-vertex model gives a $q$-deformed analogue of the Guo-Sun identity for the symmetric functions $F_{m+n-k,n}$ and $F_{n,n-k}$.
  • The appearance of the identities as commutation relations suggests that further Fehér-Némethi-Rimányi-Guo-Sun type formulas may be discovered by applying the same two-way wavefunction evaluation to other partitions.

Reading between the lines

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

  • One could test the method on other degenerations or specializations of the six-vertex model, such as flagged factorial Grothendieck polynomials, and expect analogous identities controlled by the same $D$-$B$ and $A$-$C$ commutation relations.
  • The rectangular identity likely admits a purely combinatorial proof through set-valued tableaux; the integrability argument points to which interpolation between Schur and Grothendieck versions should hold.
  • The duality formula may reflect a deeper symmetry between the $z$ and $\alpha$ variables that, once understood, would make the Guo-Sun and rectangular identities two instances of a single commutation relation.
  • The $q$-deformed identity as written is a multi-sum; simplifying it to a closed single-sum form would be a natural stress test of the integrability approach.
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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 studies identities for factorial Grothendieck polynomials at β=-1 from the viewpoint of quantum integrability. Using the established correspondence between wavefunctions of a q=0 five-vertex model and factorial Grothendieck polynomials (Eq. (2.12)), the author gives a new proof of the Guo-Sun identity (3.16), derives a new identity for rectangular Young diagrams (Theorem 4.1, Eq. (4.1)), combines it with Guo-Sun to obtain a duality formula (4.18), and sketches a q-deformed analogue leading to the identity (5.19). The proofs are based on commutation relations among D/B and A/C operators obtained from the Yang-Baxter equation.

Significance. The paper's approach is a genuine application of the algebraic Bethe ansatz to a recent combinatorial identity, and Theorem 4.1 appears to be new rather than a reproof of a known result. The proof structure is not circular, since the wavefunction-Grothendieck correspondence (2.12) is an independently published theorem and is not equivalent to the target identities. The paper also makes a potentially reusable observation: the same operator-commutation machinery works in two different directions to produce identities of Guo-Sun type. If the missing derivations are supplied, this would be a solid contribution to the interface of integrable models and K-theoretic combinatorics; small-case checks known to this reviewer are consistent with Theorem 4.1.

major comments (3)
  1. [Section 4, Eq. (4.10)] Eq. (4.10) is the sole mechanism that converts (4.4) into the factorized sum (4.14), and therefore Theorem 4.1 depends on it. The text says only that this compact commutation relation follows from (4.6)-(4.9) "by the argument in [25]", without reproducing that argument for the five-vertex representation. Because the R-matrix is the q=0 limit of the six-vertex R-matrix, the Yang-Baxter-based proof is a degenerate limit and the transfer from the phase model is not automatic. I verified the coefficients for small m with k=1, so I do not doubt the identity, but the proof as written is incomplete at a load-bearing step. Please supply a direct derivation of (4.10) from (4.6)-(4.9), for example by the coefficient-extraction argument used for (3.8).
  2. [Section 5, Eqs. (5.4), (5.14), (5.19)] The q-deformed identity (5.19) is derived after claiming that the functions F^{n,n-k} satisfy the Korepin properties (5.6)-(5.9), but the verification of those properties for the explicit functions F^{n,n-k} in (5.5) is not shown. Moreover, (5.14) is introduced as standard and its repeated application to obtain (5.15) is not demonstrated. If Section 5 is intended as a proof, these missing arguments should be supplied; otherwise the section should be explicitly labeled as a sketch or conjecture rather than a derivation.
  3. [Section 3, Eq. (3.8)] Eq. (3.8) is the key commutation relation for the Guo-Sun proof, but its derivation is again delegated to [25] with only a sketch. The sketch assumes that, when moving B-operators past D-operators, only the first terms of (3.4) contribute to the coefficient of a fixed operator ordering, and it assumes that the argument in [25] transfers to the five-vertex representation. A formal inductive proof, or a precise statement of why the representation of the quantum space is irrelevant, would make the paper self-contained. This matters because the same principle is later applied without further explanation to obtain (4.10).
minor comments (4)
  1. [Section 3, after Eq. (3.4)] In the paragraph following (3.4), "the second term of the left hand side of (3.4)" should read "the second term of the right hand side of (3.4)".
  2. [Section 4, Eq. (4.16)] The passage from (4.15) to (4.16) is not shown term by term; displaying the substitution z_j = 1-u_j^{-1}, alpha_j = 1-w_j and the resulting translations of (1-alpha_i), (z_i ⊕ alpha_j), and (alpha_j-alpha_i) would make the final identity much easier to verify.
  3. [Sections 3 and 4] Products such as prod_{i in S, j in \bar S} are written without specifying an ordering of factors. Since the relevant operators commute, this is harmless, but it is worth stating once explicitly.
  4. [Section 5, before Eq. (5.2)] The assertion that the horizontal edges between the m-th and (m+1)-th columns carry (n-k) 1s and k 0s is stated without proof; a one-sentence argument from the ice rule and the boundary conditions would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: proof reduces to independent wavefunction correspondence and R-matrix commutation, with one unproved gap that is not circular.

full rationale

The paper's derivation chain is self-contained in the sense required here. The Guo-Sun identity (1.3) is an external result that the paper re-proves by evaluating the same five-vertex wavefunction in two ways: once through the wavefunction-to-Grothendieck correspondence (2.12) and once through the operator commutation relation (3.8) plus the vacuum action (3.11). The rectangular identity (4.1) is derived analogously by comparing (4.3) and (4.14), using the commutation relation (4.10), the A-operator vacuum action (4.11), and the frozen partition function (4.13), then translating variables. None of these steps fits a parameter to the target identity, defines the polynomials in terms of the identity, or renames an input as a prediction. The correspondence (2.12) is a previously established theorem, cited to both the author's prior work and independent sources, and it does not contain the target identities. The most vulnerable step is the compact commutation relation (4.10), which is stated as following from (4.6)-(4.9) by the argument in [25] and is not fully derived; however, an unproved or under-verified lemma is a correctness gap, not circularity, and the paper does not make (4.10) true by definition or by equating it with Theorem 4.1. The q-deformation section likewise derives a new identity (5.19) from established wavefunction correspondences and standard algebraic Bethe ansatz manipulations. Accordingly, no circular step is present and the score is 0.

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

The paper adds no fitted parameters or invented entities. It rests on established correspondences and commutation relations, plus one omitted verification in the q-deformed section.

assumptions (3)
  • domain assumption Wavefunction-Grothendieck correspondence (2.12) at q=0
    Quoted from [9,10,12,13,24]; the paper does not prove it, and it is the bridge used in Sections 3-4.
  • domain assumption Commutation relations (3.8) and (4.10) for multiple D/B and A/C operators
    Quoted from [25] with a sketch; the paper relies on them to reorder operators in the proofs of (3.16) and (4.1).
  • ad hoc to paper The symmetric functions F^{n,n-k} satisfy the Korepin properties (5.6)-(5.9)
    The paper asserts 'by showing that the symmetric functions F satisfy all the properties ... we get the correspondence (5.4)' without carrying out the verification; this is an omitted proof in Section 5.

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Cite this review

Pith. "Pith review of Integrability approach to Feher-Nemethi-Rimanyi-Guo-Sun type identities for factorial Grothendieck polynomials." pith.science (2026). https://pith.science/paper/EMNJXPDN

@misc{pith2026190902278,
  author       = {Pith},
  title        = {Pith review of: Integrability approach to Feher-Nemethi-Rimanyi-Guo-Sun type identities for factorial Grothendieck polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EMNJXPDN}},
  note         = {Machine review of arXiv:1909.02278}
}
abstract

Recently, Guo and Sun derived an identity for factorial Grothendieck polynomials which is a generalization of the one for Schur polynomials by Feh\'er, N\'emethi and Rim\'anyi. We analyze the identity from the point of view of quantum integrability, based on the correspondence between the wavefunctions of a five-vertex model and the Grothendieck polynomials. We give another proof using the quantum inverse scattering method. We also apply the same idea and technique to derive an identity for factorial Grothendieck polynomials for rectangular Young diagrams. Combining with the Guo-Sun identity, we get a duality formula. We also discuss a $q$-deformation of the Guo-Sun identity.

Figures

Figures reproduced from arXiv: 1909.02278 by the authors.

Figure 1
Figure 1. The R-matrix of the six-vertex model (2.1). There are six admissible vertex con￾figurations, and the weight associated to each configuration is given below the figure of the configuration. The R-matrix (2.1) satisifies the Yang-Baxter relation Rab(u, v)Rac(u, w)Rbc(v, w) = Rbc(v, w)Rac(u, w)Rab(u, v), (2.2) acting on Wa ⊗ Wb ⊗ Wc. We denote the orthonormal basis of Wa and its dual as {|0ia, |1ia} and {ah0|, ah1|}. W… view at source ↗
Figure 2
Figure 2. The wavefunctions (2.8). The sequence of 0s on the b [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The wavefunctions corresponding to the factorial [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The wavefunction Wm+n−k,n(u1, . . . , un|w1, . . . , wm+n−k|λk + 1, . . . , λ1 + k, m + 1, . . . , m + n − k). We can see that only one configuration is allowed in the rightmost n − k columns, which gives the factor Yn j=1 u n−k j . The remaining part can be expressed …
Figure 5
Figure 5. Figure 5: The wavefunctions corresponding to the factorial [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: The partition function nh0 k 1 n−k | Y j∈Sm k C(wj |u1, . . . , un)|Ωin. We note that only one configuration is allowed, from which one can see nh0 k 1 n−k | Y j∈Sm k C(wj |u1, . . . , un)|Ωin = Yn j=1 u k j . One can easily see from the graphical description that the …
Figure 7
Figure 7. Figure 7: The decomposition of the wavefunctions (5.2). Fro [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

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