Pith. sign in

REVIEW 4 major objections 4 minor 69 references

Higher rank elliptic partition functions and multisymmetric elliptic functions

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read Exact closed forms for higher-rank elliptic partition functions

desk verdict A genuine rational-case result, but the elliptic theorem is not yet proved: the quasi-periodicity of the explicit W is never checked. read the letter →

arxiv 2412.13561 v1 pith:QE4A33GO submitted 2024-12-18 math-ph math.MP

Allan John Gerrard , Kohei Motegi , Kazumitsu Sakai This is my paper · ORCID
classification math-phmath.MP MSC 82B2317B3705E0581R12
keywords higher-rankpartitionfunctionsmultisymmetricellipticweightIzergin-KorepinanalysisnestedBetheansatzdynamicalR-matrixthetaintegrablelatticemodels
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

This paper establishes that a broad class of $\mathfrak{gl}_{M+1}$ lattice partition functions—built from $M$ layers of R-matrix vertices with extra left and right boundary quantum spaces on every level—can be written down exactly as multisymmetric functions. The authors prove the equality $\psi = W$ for the rational, trigonometric, and elliptic dynamical R-matrices (Theorems 2.3, 2.9, and 3.4). The proof uses a nested version of Izergin–Korepin analysis: a Korepin-like lemma characterizing $\psi$ by degree bounds, symmetry, recursion relations and an initial condition, followed by verification that the explicit $W$ satisfies those same properties. A sympathetic reader would care because these are exact, parameter-dependent formulas for objects that are usually only characterized implicitly, and because the elliptic formulas extend previously known weight functions used as off-shell Bethe wavefunctions.

What carries the argument

The load-bearing object is the nested Izergin–Korepin analysis—a recursion scheme that determines a partition function from structural properties—together with the extended weight functions $W$. The partition function $\psi$ is shown to satisfy a Korepin lemma: a top-layer degree bound (or, in the elliptic case, elliptic-polynomial quasi-periods), symmetry in the auxiliary variables, two recursion relations depending on whether the top right boundary colour is $M+1$, and an initial condition that reduces the top layer to a known $\mathfrak{gl}_M$ partition function. The extended weight functions $W$—nested multisymmetric sums over permutations $\sigma_1,\ldots,\sigma_M$ with rational, trigonometric, or $\theta$-function factors—are then checked against exactly these same properties, so uniqueness of the characterization yields $\psi = W$. In the elliptic case the uniqueness step rests on an elliptic interpolation theorem (Proposition 3.1), which identifies an elliptic polynomial of degree $k_M$ from its values at $k_M$ points.

What would settle it

Evaluate both sides of (38) for the smallest case with left and right boundary sites, say $M=2$, $L^I_1 = L^{II}_1 = 1$, $k_2=1$, at a generic point not among the interpolation values; any mismatch in the values, or any failure of the weight function $W$ to satisfy the quasi-periodicity (31) under $w^{(M)}_{L_M} \to w^{(M)}_{L_M} + \tau$, would disprove the elliptic theorem.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the partition function $\psi$ defined graphically from the R-matrix is not merely characterized recursively but is identically equal to the extended weight function $W$: $\psi = W$ for every configuration allowed by the labelling of the paper, including configurations with nonempty left and right boundary sites $y^{(j)}_I, y^{(j)}_{II}$ at each intermediate level. Here $W$ is an explicit multisymmetric function given as a nested sum over permutations of the auxiliary spectral variables, with factors built from the rational, trigonometric, or elliptic R-matrix weights. In the elliptic case the identity is proved with the help of an elliptic Lagrange interpolation uniqueness statement, and in the boundary-free special case the elliptic $W$ reduces to the previously known elliptic weight functions. The paper thus presents a unified treatment of all three R-matrix types.

Load-bearing premise

The proof's load-bearing premise is uniqueness: the stated degree bound (or elliptic quasi-periods), symmetry, recursion relations and initial condition must single out exactly one function, and in the elliptic case the quasi-periods (30)–(31) must be exactly right for the interpolation theorem to apply.

Editorial extensions

If this is right

  • With $\psi = W$ established, every partition function in this family has a closed multisymmetric expression, so off-shell nested Bethe wavefunctions can be studied by manipulating $W$ directly.
  • Setting all left and right boundary sets empty recovers the original partition functions of the prior work as a special case, so the new formulas strictly generalize that construction.
  • For the elliptic case with no intermediate boundary sites, $W$ reduces to the previously known elliptic weight functions, giving a lattice-model derivation of those special functions.
  • The same nested Korepin-lemma scheme works uniformly for the rational, trigonometric and elliptic R-matrices, so one proof template covers all three regimes.

Reading between the lines

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

  • One testable consequence left implicit: because $W$ is explicitly multisymmetric, one could attempt to derive determinant or contour-integral representations for these partition functions, in analogy with scalar-product formulas; the paper does not carry that out.
  • The elliptic quasi-periods (30)–(31) are proved for $\psi$ but not independently checked for $W$; a symbolic verification of those exact quasi-periods for $W$ would close the last gap in the elliptic uniqueness argument.
  • The conclusion's suggestion of supersymmetric analogues gives a direct route: re-running the nested Korepin lemma with supersymmetric R-matrices should produce $\mathfrak{gl}_{M+1|N}$ weight functions, though that is not done here.
  • The possible link to stable bases and quiver varieties, mentioned as future work, would let these partition functions serve as explicit formulas for stable-envelope classes beyond the boundary-free cases.
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

4 major / 4 minor

Summary. The paper introduces a family of gl_{M+1} lattice partition functions that extend the Foda-Manabe construction by adding left and right boundary quantum spaces at every level. For the rational, trigonometric, and elliptic R-matrices, the authors state a nested Korepin lemma and use it to characterize the partition function, then propose explicit extended multisymmetric weight functions W and claim ψ = W (Theorems 2.3, 2.9, and 3.4). The rational case is supported by detailed computations in Propositions 2.4–2.6; the trigonometric case is stated to follow by the same strategy; the elliptic case is summarized with a brief proof sketch. Section 4 specializes the elliptic formulas and compares them with elliptic weight functions of Konno and Rimányi–Tarasov–Varchenko.

Significance. If the main identities hold, the paper gives explicit multisymmetric formulas for a substantially enlarged class of gl_{M+1} partition functions and unifies the rational, trigonometric, and elliptic weight-function families. The rational proof is concrete and the special-case comparison in Section 4 is a useful contribution. The elliptic claim is, however, not currently established: the essential quasi-periodicity check for the candidate function W is missing, and the manuscript also contains a discrepancy in the displayed elliptic recursion. These are localized and fixable issues, but they are load-bearing for the central theorem, so I cannot recommend acceptance in the present form.

major comments (4)
  1. [§3, Theorem 3.4] The elliptic uniqueness argument is incomplete. Proposition 3.1 can be applied only if both the partition function and the explicit candidate function W lie in Θ_{k_M}(χ) with the same quasi-periods. Proposition 3.2 establishes the quasi-periods (30)–(31) for ψ, but the proof of Theorem 3.4 says only that the proof is 'the same as the rational/trigonometric case' and then lists relations among the C^{(p)} symbols for Properties 3–5. No verification is given that the function defined by (37), as a function of w^{(M)}_{L_M}, satisfies (30)–(31). This is not a cosmetic omission: the interpolation step uses exactly k_M point evaluations, and a different γ-shift in the character would make Proposition 3.1 inapplicable. Please add the explicit computation of W(w^{(M)}_{L_M}+1) and W(w^{(M)}_{L_M}+τ) from (37), or give a self-contained induction proving the quasi-periodicity.
  2. [§3, Proposition 3.2, Eq. (32)] The displayed recursion coefficient in (32) does not match the derivation in the proof. The proof ends with the factor -[Λ_{M+1}-Λ_i+γ][γ] / [Λ_{M+1}-Λ_i-(k_M-L_M)γ], which is equal to [γ][λ_{M+1}-λ_i+γ(k_M-L_M+C^{(M)}(L_I,i))] / [λ_i-λ_{M+1}+γ(1-C^{(M)}(L_I,i))]. Equation (32) instead writes [γ] divided by the product of these two theta brackets. As written, the printed recursion differs from the derived one by a factor of [λ_{M+1}-λ_i+γ(k_M-L_M+C^{(M)}(L_I,i))]^2. Since this recursion is part of the data used in the uniqueness argument, the formula must be corrected or an explanation must be given for the missing cancellation.
  3. [§3, Definition 3.3 and Theorem 3.4] The notation in the elliptic case is inconsistent. Definition 3.3 presents formula (37) with the symbol ψ on the left-hand side and calls it the extended elliptic weight function, while Theorem 3.4 states ψ = W. As printed, W is never defined for the elliptic case, so the theorem is ill-posed or tautological. Presumably (37) is intended to define W; please fix the notation consistently throughout Section 3 and in the statement of Theorem 3.4.
  4. [§2.5, Theorem 2.9] The trigonometric theorem is not actually proved in the text. The statement that the strategy is identical to the rational case is not a proof, because the q-dependent R-matrix (14) changes the weights and the specialization point v^{(M)}_{L_M}=q^{-1}u^{(M)}_{k_M} in Property 3 of Proposition 2.7. The analogues of Propositions 2.4–2.6 for the function (19) are not stated. I recommend either providing the trigonometric recursion proofs or giving a precise reduction showing that the rational computations apply verbatim to (14)–(19).
minor comments (4)
  1. [§3, Proposition 3.1] The interpolation condition after the points y_j is misprinted: 'P k y_k − α' should presumably be '∑_k y_k − α ∉ Γ'.
  2. [§3, proof of Proposition 3.2] In the proof of Property 4 or 5 there is a typo 'L_I^{M1}' in the sentence about I^{(M)}_{k_M}; it should be 'L_I^{M-1}' or similar.
  3. [§2, proof of Proposition 2.4] The displayed equations in Step 1 contain strikethrough/cancelled factors; the camera-ready version should remove these editorial marks.
  4. [§4] The equivalence of (40) with the Konno and Rimányi–Tarasov–Varchenko formulas is shown by a dictionary between symbols and a statement that the expressions are equivalent; please add at least a short explanation of why the unordered multisets of summands and the theta arguments coincide after the stated relabellings.

Circularity Check

0 steps flagged · score 1.0 of 10

No substantive circularity: the identity ψ = W is proved through independent Korepin properties; only a notation collision and an omitted elliptic quasi-period check are flagged as non-circular caveats.

full rationale

The paper's central identity ψ = W is not circular. The partition function ψ is defined from the lattice R-matrix diagram, and the candidate W is defined independently as a multisymmetric sum over permutations (Definition 2.2). Theorem 2.3 is proved by checking that W satisfies exactly the nested Korepin characterization of ψ from Proposition 2.1: the degree bound and x^{(M)}-symmetry are immediate, and Propositions 2.4–2.6 verify the two recursion relations and the initial condition; these properties determine the partition function by the nested induction described in §2.3. Thus the equality does not assume its conclusion. The trigonometric case is the same argument in u/v variables, and the elliptic case is intended to repeat it. No parameter is fitted to data, and no prediction is a fitted input renamed. The only self-citations ([30], [33], [69]) concern earlier special cases or the authors' prior Bethe-ansatz work; the elliptic interpolation uniqueness theorem (Proposition 3.1) is an external Felder–Schorr result, not an imported self-theorem, so the elliptic uniqueness step does not reduce to a self-citation. Two caveats are worth flagging, though neither is circularity: (i) Theorem 3.4 is proved only by the sentence 'The proof is the same as the rational/trigonometric case', and the manuscript does not explicitly verify that the explicit elliptic function (37) lies in Θ_{k_M}(χ) with the quasi-periods (30)–(31) required by Proposition 3.1; this is an omitted check, not a definitional identification. (ii) In Definition 3.3 the explicit function is printed with the symbol ψ, while Theorem 3.4 prints ψ = W, so taken literally the elliptic statement is self-referential; the surrounding text and Section 4 make the intended reading unambiguous. These are correctness/completeness or typographical issues, not evidence that the derivation reduces to its inputs.

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

The paper introduces no free parameters fitted to data and no invented physical entities. The 'colors', Foda-Manabe labeling, and C^{(p)} counting functions are combinatorial bookkeeping, not new physical objects. The central claim rests on standard background from integrable systems (Yang-Baxter equation, ice rule) and an external interpolation theorem, all properly cited.

assumptions (4)
  • domain assumption The rational, trigonometric, and elliptic R-matrices satisfy the Yang-Baxter equation (2), (15), and the dynamical Yang-Baxter equation (25).
    Used to prove symmetry in the top spectral variable (Property 2) via the train argument; the elliptic version requires the dynamical shifts as stated in Section 3.
  • standard math Elliptic Lagrange interpolation theorem (Proposition 3.1, cited from [59]).
    Underpins the uniqueness part of the elliptic Izergin-Korepin argument in Section 3; requires the function to lie in Θ_{k_M}(χ) with specific quasi-periods.
  • domain assumption Ice rule (color conservation) at each vertex of the lattice model.
    Justifies the counting |I^{(j)}_{k_j}| = k_j and the recursion structure in Section 2.2; the elliptic R-matrix is assumed to preserve this rule.
  • domain assumption The partition function is well-defined for arbitrary orderings of auxiliary spectral variables due to Yang-Baxter.
    Stated in Section 2.2 and used throughout the train argument; relies on standard properties of integrable vertex models.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Higher rank elliptic partition functions and multisymmetric elliptic functions." pith.science (2026). https://pith.science/paper/QE4A33GO

@misc{pith2026241213561,
  author       = {Pith},
  title        = {Pith review of: Higher rank elliptic partition functions and multisymmetric elliptic functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QE4A33GO}},
  note         = {Machine review of arXiv:2412.13561}
}
abstract

We introduce and investigate a class of $\mathfrak{gl}_{M+1}$ partition functions which is an extension of the one introduced by Foda-Manabe. We characterize the partition functions by a nested version of Izergin-Korepin analysis, and determine the explicit forms, for each of the rational, trigonometric and elliptic versions. The resulting multisymmetric functions can be regarded as extensions of the rational, trigonometric and elliptic weight functions.

Figures

Figures reproduced from arXiv: 2412.13561 by the authors.

Figure 1
Figure 1. Matrix elements of the rational R-matrix R(x, y). The R-matrix satisfies the Yang-Baxter equation R23(x2 − x3)R13(x1 − x3)R12(x1 − x2) = R12(x1 − x2)R13(x1 − x3)R23(x2 − x3), (2) acting on V1 ⊗ V2 ⊗ V3. Each Rij (x, y) acts nontrivially on Vi and Vj , and on the remaining space as identity, for example R12(x, y) = R(x, y) ⊗ I. See [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The Yang-Baxter equation (2). x (j) = {x (j) 1 , . . . , x (j) kj } (kj = |x (j) |). Note that the ordering of the variables in the auxiliary spaces can be arbitrary due to the Yang-Baxter relation. The spectral variables in the quantum spaces in the j-th layer are y (j) I = {y (j) I,1 , . . . , y (j) I,LI j } (L I j = |y (j) I |), x (j+1) , y (j) II = {y (j) II,1 , . . . , y (j) II,LII j } (L II j = |y (j) II |). N… view at source ↗
Figure 3
Figure 3. Rational and trigonometric glM+1 partition functions (4) and (13). For the trigonometric case, the x- and y-variables are replaced by u- and v-variables. and output must be identical for each quantum space; in other words, all outputs at layer j are of colour j. Then, for each auxiliary space that has output of colour j, the number of outputs that must be of colour j is reduced by one. The set of coordinates that ar… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Pictorial explanation of the induced label. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: Sets used to describe the configurations. On the left panel, each row corresponds to each layer of the [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: Pictorial explanation of property 3 of the partition function. At this value of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Evaluation of the weights from the frozen nodes. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Pictorial explanation for Case B. 2.4 Multisymmetric functions In this subsection, we introduce a class of multisymmetric functions and show they are explicit forms of the partition functions for the rational case. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: The case when kM = 1 and i (M) LI = j ̸= M + 1. The top layer is frozen, and removing that part gives partition function of glM. Definition 2.2. We define the extended rational weight function as W  x (1) , . . . , x (M) [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: Matrix elements of the trigonometric R-matrix R(u, v). Proposition 2.7. Let L I := L I 1 + · · · + L I M−1 + LM. The functions ψ satisfy the following properties: 1. Case A: If i (M) LI ̸= M + 1, the degree of v (M) LM in ψ is at most kM − 1. 2. ψ is symmetric with re…
Figure 11
Figure 11. Figure 11: The matrix elements of the dynamical R-matrix. Here λ ∈ h ∗ is the dynamical parameter which can be viewed as λ ∈ C n by the expansion λ = Pn i=1 λiµi . The dynamical parameter associated to an R-matrix is the one that is in the upper left corner of the plaquet. Then,…
Figure 12
Figure 12. Figure 12: The dynamical Yang-Baxter equation (25) or equivalently the star-triangle relation. [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: Simplified drawing of the dynamical R-matrix. some corner changes if one crosses a line and move to another corner of the plaquet, and the parameter associated with the northwest corner in the j-th layer (counted from bottom) becomes λ [j] . Finally, we introduce the …
Figure 14
Figure 14. Figure 14: The elliptic partition function. 4. Case B: If i (M) LI = M + 1, ψ  z (1) , . . . , z (M) [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: The symbol C (p) (k, l) which may be explained diagrammatically as follows. Among the sites on level p or greater, but with site number no greater than k (highlighted in blue), count the number of sites coloured by l. 5. Initial condition: If kM = 1 and i (M) LI ̸= M …
Figure 16
Figure 16. Figure 16: The evaluation of the rightmost column of the partition function [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]
Figure 17
Figure 17. Figure 17: Diagram describing the evaluation of the elliptic partition function for Case A. [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

69 extracted references · 67 canonical work pages

  1. [1]

    Bethe, Z

    H. Bethe, Z. Phys. 71, 205 (1931)

    1931

  2. [2]

    Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982)

    R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982)

    work page 1982

  3. [3]

    Korepin, N.M

    V.E. Korepin, N.M. Bogoliubov and A.G. Izergin Quantum Inverse Scattering Method and Correlation functions (Cambridge University Press, Cambridge, 1993)

    work page 1993

  4. [4]

    Korepin, Commun

    V.E. Korepin, Commun. Math. Phys. 86, 391 (1982). 28

    work page 1982

  5. [5]

    Izergin Sov

    A. Izergin Sov. Phys. Dokl. 32, 878 (1987)

    work page 1987

  6. [6]

    Kuperberg, Int

    G. Kuperberg, Int. Math. Res. Not. 3, 123 (1996)

    work page 1996

  7. [7]

    Kuperberg, Ann

    G. Kuperberg, Ann. Math. 156, 835 (2002)

    work page 2002

  8. [8]

    Tsuchiya, J

    O. Tsuchiya, J. Math. Phys. 39, 5946 (1998)

    work page 1998

Show all 69 references
  1. [9]

    Brubaker, D

    B. Brubaker, D. Bump and S. Friedberg, Commun. Math. Phys. 308, 281 (2011)

    2011

  2. [10]

    Tarasov and A

    V. Tarasov and A. Varchenko, SIGMA 9, 048 (2013)

    2013

  3. [11]

    Betea, M

    D. Betea, M. Wheeler and P. Zinn-Justin, J. Alg. Comb. 42, 555 (2015)

    2015

  4. [12]

    Borodin and L

    A. Borodin and L. Petrov, Sel. Math. New Ser. 24 751 (2016)

    2016

  5. [13]

    van Diejen and E

    J.F. van Diejen and E. Emsiz, Commun. Math. Phys. 350, 1017 (2017)

    2017

  6. [14]

    Borodin, Adv

    A. Borodin, Adv. in Math. 306, 973 (2017)

    2017

  7. [15]

    Takeyama, Funkcialaj Ekvacioj, 61, 349 (2018)

    Y. Takeyama, Funkcialaj Ekvacioj, 61, 349 (2018)

    2018

  8. [16]

    Brubaker, V

    B. Brubaker, V. Buciumas, D. Bump and N. Gray, Comm. Numb. Theor. Phys. 13, 101 (2019)

    2019

  9. [17]

    Foda and M

    O. Foda and M. Manabe, J. High Energ. Phys. 2019, 36 (2019)

    2019

  10. [18]

    Borodin and M

    A. Borodin and M. Wheeler, Ast´ erisque, 437 (2022)

    2022

  11. [19]

    Brubaker, C

    B. Brubaker, C. Frechette, A. Hardt, E. Tibor and K. Weber, Alg. Comb. 6, 789 (2023)

    2023

  12. [20]

    Felder and A

    G. Felder and A. Varchenko, Nucl. Phys. B, 480, 485 (1996)

    1996

  13. [21]

    Tarasov and A

    V. Tarasov and A. Varchenko, Ast´ erisque,246 (1997)

    1997

  14. [22]

    Pakuliak, V

    S. Pakuliak, V. Rubtsov and A. Silantyev, J. Phys. A:Math. Theor. 41, 295204 (2008)

    2008

  15. [23]

    Rosengren, Adv

    H. Rosengren, Adv. Appl. Math. 43, 137 (2009)

    2009

  16. [24]

    Filali and N

    F. Filali and N. Kitanine, J. Stat. Mech. L06001 (2010)

    2010

  17. [25]

    W-L. Yang, X. Chen, J. Feng, K. Hao, K-J. Shi, C-Y. Sun, Z-Y. Yang and Y-Z. Zhang, Nucl. Phys. B 847, 367 (2011)

    2011

  18. [26]

    W-L. Yang, X. Chen, J. Feng, K. Hao, K. Wu, Z-Y. Yang and Y-Z. Zhang, Nucl. Phys. B 848, 523 (2011)

    2011

  19. [27]

    W.Galleas, Nucl. Phys. B 858, 117 (2012)

    2012

  20. [28]

    Galleas, J

    W. Galleas, J. Lamers, Nucl. Phys. B 886, 1003 (2014)

    2014

  21. [29]

    Lamers, Nucl

    J. Lamers, Nucl. Phys. B 901, 556 (2015)

    2015

  22. [30]

    Motegi, J

    K. Motegi, J. Math. Phys. 59, 053505 (2018)

    2018

  23. [31]

    Aggarwal, Sel

    A. Aggarwal, Sel. Math. New Ser. 24, 2659 (2018)

    2018

  24. [32]

    Borodin, J

    A. Borodin, J. Eur. Math. Soc. 22, 1353 (2020)

    2020

  25. [33]

    Motegi, J

    K. Motegi, J. Math. Phys. 61, 053507 (2020)

    2020

  26. [34]

    Matsuo, Comm

    A. Matsuo, Comm. Math. Phys. 157, 479 (1993)

    1993

  27. [35]

    Tarasov and A

    V. Tarasov and A. Varchenko, Leningrad Math. J. 6, 275 (1994)

    1994

  28. [36]

    Mimachi, Duke Math.J

    K. Mimachi, Duke Math.J. 85, 635 (1996). 29

    1996

  29. [37]

    Rim´ anyi, V

    R. Rim´ anyi, V. Tarasov and A. Varchenko, J. Geom. Phys.94, 81 (2015)

    2015

  30. [38]

    Shenfeld, Abelianization of Stable Envelopes in Symplectic Resolutions, PhD thesis, Princeton, 2013

    D. Shenfeld, Abelianization of Stable Envelopes in Symplectic Resolutions, PhD thesis, Princeton, 2013

    2013

  31. [39]

    Kosmakov, V

    M. Kosmakov, V. Tarasov, arXiv:2312.00980

    arXiv

  32. [40]

    Kosmakov, V

    M. Kosmakov, V. Tarasov, arXiv:2402.15717

    arXiv

  33. [41]

    N. Yu. Reshetikhin, J. Sov. Math. 46, 1694 (1989)

    1989

  34. [42]

    Smirnov, Sel

    A. Smirnov, Sel. Math. New Ser. 26, 1 (2020)

    2020

  35. [43]

    Konno, J

    H. Konno, J. Int. Syst. 2, xyx011 (2017)

    2017

  36. [44]

    Konno, J

    H. Konno, J. Int. Syst. 3, xyy012 (2018)

    2018

  37. [45]

    Felder, R

    G. Felder, R. Rim´ anyi and A. Varchenko, SIGMA14, 41 (2018)

    2018

  38. [46]

    Rim´ anyi, V

    R. Rim´ anyi, V. Tarasov and A. Varchenko, Sel. Math.25, 16 (2019)

    2019

  39. [47]

    Aganagic and A

    M. Aganagic and A. Okounkov, J. Amer. Math. Soc. 34, 79 (2021)

    2021

  40. [48]

    Maulik and A

    D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, Ast´ erisque, 408 (2019)

    2019

  41. [49]

    Nekrasov and S

    N. Nekrasov and S. Shatashvili, Nucl. Phys. Proc. Supp. 192-193, 91 (2009)

    2009

  42. [50]

    Nekrasov and S

    N. Nekrasov and S. Shatashvili, Prog. Theor. Phys. Supp. 177, 105 (2009)

    2009

  43. [51]

    Wheeler, Nucl

    M. Wheeler, Nucl. Phys. B 852, 468 (2011)

    2011

  44. [52]

    Slavnov, Theor

    N.A. Slavnov, Theor. Math. Phys. 79, 502 (1989)

    1989

  45. [53]

    V. G. Drinfeld, Soviet Math. Dokl. 36, 212 (1988)

    1988

  46. [54]

    M. Jimbo. Lett. Math. Phys. 10, 63 (1985)

    1985

  47. [55]

    Reshetikhin, L.A

    N.Y. Reshetikhin, L.A. Takhtadzhyan and L.D. Faddeev. Algebra i Analiz 1, 178 (1989)

    1989

  48. [56]

    Felder, Elliptic quantum groups

    G. Felder, Elliptic quantum groups. In: Iagolnitzer, D. (ed.) Proceedings of the ICMP, Paris 1994, pp. 211-218. Intern. Press, Cambridge, MA (1995)

    1995

  49. [57]

    Felder and A

    G. Felder and A. Varchenko, Comm. Math. Phys. 181, 741 (1996)

    1996

  50. [58]

    Cavalli, On representations of the Elliptic Quantum Group Eγ,τ (glN ), PhD thesis, 2001, ETH Z¨ urich

    A. Cavalli, On representations of the Elliptic Quantum Group Eγ,τ (glN ), PhD thesis, 2001, ETH Z¨ urich

    2001

  51. [59]

    Felder and A

    G. Felder and A. Schorr, J. Phys. A: Math.Gen. 32, 8001 (1999)

    1999

  52. [60]

    Andrews, R.J

    G.E. Andrews, R.J. Baxter and P.J. Forrester, J. Stat. Phys. 35, 193 (1984)

    1984

  53. [61]

    Baxter, Ann

    R.J. Baxter, Ann. Phys. 70, 193 (1972)

    1972

  54. [62]

    O. Foda, K. Iohara, M. Jimbo, R. Kedem, T. Miwa and H. Yan, Lett. Math. Phys. 32, 259 (1994)

    1994

  55. [63]

    Fronsdal, Lett

    C. Fronsdal, Lett. Math. Phys. 40, 117 (1997)

    1997

  56. [64]

    Konno, Comm

    H. Konno, Comm. Math. Phys. 195, 373 (1998)

    1998

  57. [65]

    Jimbo, H

    M. Jimbo, H. Konno, S. Odake and J. Shiraishi, Trans. Groups. 4, 303 (1999)

    1999

  58. [66]

    E. Date, M. Jimbo, A. Kuniba, T. Miwa and M. Okado, Nucl. Phys.B 290, 231 (1987)

    1987

  59. [67]

    Jimbo, A

    M. Jimbo, A. Kuniba, T. Miwa and M. Okado, Comm. Math. Phys. 119, 543 (1988)

    1988

  60. [68]

    Liashyk and S

    A. Liashyk and S. Z. Pakuliak. ‘Algebraic Bethe Ansatz for o2n+1-Invariant Integrable Models’. Theor. and Math. Phys. 206, no. 1 (2021): 19–39

    2021

  61. [69]

    Gerrard and V

    A. Gerrard and V. Regelskis. ‘Nested Algebraic Bethe Ansatz for Deformed Orthogonal and Symplectic Spin Chains’. Nucl. Phys. B 956C. (2020) 30

    2020

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.