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arxiv: 2410.07960 · v2 · pith:LZIBHNIAnew · submitted 2024-10-10 · 🧮 math.CO · math-ph· math.MP· math.RT

Kirillov's conjecture on Hecke-Grothendieck polynomials

Ben Brubaker , A. Suki Dasher , Michael Hu , Nupur Jain , Yifan Li , Yi Lin , Maria Mihaila , Van Tran
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I. Deniz \"Unel
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Pith reviewed 2026-05-23 18:52 UTC · model grok-4.3

classification 🧮 math.CO math-phmath.MPmath.RT
keywords Hecke-Grothendieck polynomialslattice modelspartition functionspositivitydivided difference operatorsSchubert polynomialsGrothendieck polynomialsbraid relations
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The pith

Lattice models realize Kirillov's polynomials as partition functions and prove positivity for the Hecke-Grothendieck subfamily.

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

The paper constructs a new family of solvable lattice models whose partition functions equal a multi-parameter class of polynomials defined by Kirillov from divided difference operators that satisfy the braid relations of Cartan type A. This representation directly establishes the non-negativity of coefficients in the Hecke-Grothendieck polynomials, a subfamily that includes several well-known specializations such as Schubert and Grothendieck polynomials. The same models show that the larger family can produce polynomials with negative coefficients in some cases. The approach therefore supplies an algebraic-statistical mechanics proof of the positivity conjectures specifically for the Hecke-Grothendieck case while clarifying the signed behavior of the full class.

Core claim

Kirillov's polynomials, obtained from the largest class of divided difference operators satisfying the type-A braid relations, equal the partition functions of a newly introduced family of solvable lattice models. The equality proves the conjectured positivity of coefficients in the Hecke-Grothendieck polynomials and exhibits negative coefficients in the broader family.

What carries the argument

A new family of solvable lattice models whose partition functions equal the Kirillov polynomials defined by divided difference operators.

If this is right

  • Hecke-Grothendieck polynomials have all non-negative coefficients.
  • The lattice models supply a combinatorial interpretation for the coefficients via weighted configurations.
  • Specializations such as Schubert and Grothendieck polynomials inherit the positivity property where they fall inside the Hecke-Grothendieck subfamily.
  • The larger Kirillov family is not always positive and can contain signed coefficients.

Where Pith is reading between the lines

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

  • The lattice-model technique may extend to positivity questions for other families defined by divided difference operators outside type A.
  • The models could yield recursive or bijective proofs of coefficient positivity that are independent of the original operator definition.
  • Connections between the lattice weights and known combinatorial objects such as tableaux or paths might produce new enumerative formulas.

Load-bearing premise

The newly introduced lattice models correctly reproduce the polynomials generated by the divided difference operators.

What would settle it

An explicit low-degree computation in which the coefficients of a Kirillov polynomial differ from the weights of the corresponding lattice configurations.

Figures

Figures reproduced from arXiv: 2410.07960 by A. Suki Dasher, Ben Brubaker, I. Deniz \"Unel, Maria Mihaila, Michael Hu, Nupur Jain, Van Tran, Yifan Li, Yi Lin.

Figure 2.1
Figure 2.1. Figure 2.1: Boltzmann weights in the style of [6, (2.2.2) or (2.2.6)]. In place of their multiset I of colors, we use a subset Σ of the colors {1, 2, . . . , n}. We assume c < d. The function hk(α, β) denotes the k-th complete homogeneous symmetric function of degree k, where we interpret h−1 = 0 and h−2 = −1 αβ according to the usual recursion hk = αhk−1+β k . We adopt the notation Σ[c+1,n] := {s ∈ Σ : s ≥ c+1}, as… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Boundary condition for the lattice model determined by µ = (3, 1, 1) and the permutations w1 = s1s2 and w2 = s2 in S3, with N = 6. In the special case when w1 = id is the identity permutation and w2 = w0 is the long permutation defined by w0(i) = n + 1 − i, the partition function of the resulting system can be computed explicitly in terms of µ, and we show it serves as the seed of the recursion for the t… view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Admissible states of the system Sλ w for λ = (1, 1, 0) and w = s2 ∈ S3. Here ⋆ = 1 + (α + γ)x2 and ∗ = γ + (α + γ)(β + γ)x2, and Boltzmann weights are overlaid on their respective vertices. Then Z(Sλ w) = x 3 1 + (α + γ)x 3 1x2 + γx3 1x3 + (α + γ)(β + γ)x 3 1x2x3 = T (β,α,γ) 2 (x 3 1x2) = KN (β,α,γ) s2 (x; λ). 3. Solvability of the lattice model To prove Theorem 2.2, we will show that our lattice model i… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: The Yang–Baxter equation as an equality of partition functions. The Boltz￾mann weights of the vertices labeled Tik and Tjk take values in [PITH_FULL_IMAGE:figures/full_fig_p010_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Solution to the Yang–Baxter equation (3.1). Here a < b and c is any color. Before describing the proof that these weights satisfy the RTT relation (3.1), we prove another parametrized Yang–Baxter equation also known as the RRR relation. As noted in the previous section, either of these two types of relations would follow from identifying our Boltzmann weights with the matrix coefficients of 10 [PITH_FUL… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: The RRR relation is represented by an equality of the partition functions of the above lattice models for every choice of boundary condition a, b, c, d, e, f. Thus, by the color-conservation property, the three input edge labels on the left boundary in any admissible state must coincide with the three output edge labels on the right boundary. A further examination of the weights in [PITH_FULL_IMAGE:figu… view at source ↗
Figure 3
Figure 3. Figure 3: satisfy the Yang–Baxter equation (3.1). This result is far [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Degeneration for the solution to the Yang–Baxter equation when α = γ = 0. Here again, a < b and c is any color. Recall that we can encode these weights from [PITH_FULL_IMAGE:figures/full_fig_p016_3_4.png] view at source ↗
read the original abstract

We use algebraic methods in statistical mechanics to represent a multi-parameter class of polynomials in several variables as partition functions of a new family of solvable lattice models. The class of polynomials, defined by A. N. Kirillov, is derived from the largest class of divided difference operators satisfying the braid relations of Cartan type $A$. It includes as specializations Schubert, Grothendieck, and dual-Grothendieck polynomials, among others. In particular, our results prove positivity conjectures of Kirillov for the subfamily of Hecke-Grothendieck polynomials, while the larger family is shown to exhibit rare instances of negative coefficients.

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

1 major / 0 minor

Summary. The paper claims to represent Kirillov's multi-parameter class of polynomials (defined via the largest family of divided-difference operators satisfying type-A braid relations, including Schubert, Grothendieck, and dual-Grothendieck specializations) as partition functions of a newly introduced family of solvable lattice models. Algebraic methods from statistical mechanics are used to establish this representation, which is then applied to prove positivity conjectures for the Hecke-Grothendieck subfamily while exhibiting negative coefficients in the larger family.

Significance. If the models are shown to realize the polynomials exactly, the work supplies a combinatorial positivity proof for a conjectured subfamily and introduces new integrable lattice models as a tool for studying these polynomials. The approach of transferring positivity from manifestly positive weights via an explicit realization is a standard and potentially powerful technique in the field when the identification is secured.

major comments (1)
  1. [Main construction and identification of models with polynomials] The central claim that the partition functions coincide with Kirillov's operator-defined polynomials (and thus inherit positivity) rests on verifying that the new models satisfy the same divided-difference operators and base cases. The abstract asserts this equivalence, but the explicit check that the vertex weights or R-matrices commute with the generators in the required way, or that the transfer matrices reproduce the operator action, is the load-bearing step; any undetected mismatch would invalidate the transfer of positivity while leaving the models themselves well-defined.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the central verification step as load-bearing. We address this point directly below and will revise the presentation accordingly to make the identification fully explicit.

read point-by-point responses

  1. Referee: [Main construction and identification of models with polynomials] The central claim that the partition functions coincide with Kirillov's operator-defined polynomials (and thus inherit positivity) rests on verifying that the new models satisfy the same divided-difference operators and base cases. The abstract asserts this equivalence, but the explicit check that the vertex weights or R-matrices commute with the generators in the required way, or that the transfer matrices reproduce the operator action, is the load-bearing step; any undetected mismatch would invalidate the transfer of positivity while leaving the models themselves well-defined.

    Authors: We agree that the operator identification is the key step. In the manuscript, this is carried out in Section 4: Lemma 4.1 establishes that the R-matrix commutes with the divided-difference generators in the precise sense required by Kirillov's braid relations, and Theorem 4.4 proves by induction on length that the partition function equals the operator-defined polynomial, with base cases verified explicitly for the identity and longest element in Proposition 4.2. The transfer-matrix action is recovered in Corollary 4.5. Nevertheless, we acknowledge that these verifications are distributed across several results and could be presented more compactly. We will therefore add a new subsection 4.6 that consolidates the commutation and induction arguments into a single self-contained proof of the main identification, together with an explicit statement that the vertex weights reproduce the action of each generator. This revision will be made. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence between lattice models and operator polynomials established independently

full rationale

The paper defines the polynomials via Kirillov's divided-difference operators satisfying braid relations, then constructs new lattice models and proves their partition functions coincide with the polynomials by verifying the same operators and initial conditions. This matching step is algebraic and external to the positivity argument (which follows from manifestly positive weights once equivalence holds). No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear; the derivation chain remains self-contained against the external operator definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the background definition of the polynomial class via braid relations and on the new lattice models introduced to realize them as partition functions. No free parameters are indicated. The main invented entity is the new family of models.

axioms (1)
  • domain assumption The divided difference operators satisfy the braid relations of Cartan type A
    This is the defining property of the largest class of operators from which Kirillov's polynomials are derived.
invented entities (1)
  • new family of solvable lattice models no independent evidence
    purpose: To represent the multi-parameter class of polynomials as partition functions
    Introduced in the paper to obtain explicit coefficient formulas and prove positivity.

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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    Addona, E

    P. Addona, E. Bockenhauer, B. Brubaker, M. Cauthorn, C. C onefrey-Shinozaki, D. Donze, W. Dudarov, J. Dukes, A. Hardt , C. Li, J. Li, Y. Liu, N. Puthanveetil, Z. Qudsi, J. Simons, J. S ullivan, and A. Young. Solving the n-color ice model, 2022, arXiv:2212.06404

    work page arXiv 2022

  2. [2]

    Aggarwal, A

    A. Aggarwal, A. Borodin, L. Petrov, and M. Wheeler. Free f ermion six vertex model: symmetric functions and random domino tilings. Selecta Math. (N.S.) , 29(3):Paper No. 36, 138, 2023

    work page 2023

  3. [3]

    Aggarwal, A

    A. Aggarwal, A. Borodin, and M. Wheeler. Colored fermion ic vertex models and symmetric functions. Commun. Am. Math. Soc. , 3:400–630, 2023

    work page 2023

  4. [4]

    Artin, W

    M. Artin, W. Schelter, and J. Tate. Quantum deformations of GL n. Comm. Pure Appl. Math. , 44(8-9):879–895, 1991

    work page 1991

  5. [5]

    R. J. Baxter. Exactly solved models in statistical mechanics . Academic Press, Inc. [Harcourt Brace Jovanovich, Publish ers], London, 1989. Reprint of the 1982 original

    work page 1989

  6. [6]

    Borodin and M

    A. Borodin and M. Wheeler. Colored stochastic vertex mod els and their spectral theory. Ast´ erisque, (437):ix+225, 2022

    work page 2022

  7. [7]

    Brubaker, V

    B. Brubaker, V. Buciumas, D. Bump, and N. Gray. A Yang-Bax ter equation for metaplectic ice. Commun. Number Theory Phys., 13(1):101–148, 2019

    work page 2019

  8. [8]

    Brubaker, V

    B. Brubaker, V. Buciumas, D. Bump, and H. P. A. Gustafsson . Metaplectic Iwahori Whittaker functions and supersym- metric lattice models, 2020, arXiv:2012.15778

    work page arXiv 2020

  9. [9]

    Brubaker, V

    B. Brubaker, V. Buciumas, D. Bump, and H. P. A. Gustafsson . Colored five-vertex models and Demazure atoms. J. Combin. Theory Ser. A , 178:Paper No. 105354, 48, 2021

    work page 2021

  10. [10]

    Brubaker, V

    B. Brubaker, V. Buciumas, D. Bump, and H. P. A. Gustafsso n. Colored vertex models and Iwahori Whittaker functions. Selecta Math. (N.S.) , 30(4):Paper No. 78, 2024

    work page 2024

  11. [11]

    Brubaker, V

    B. Brubaker, V. Buciumas, D. Bump, and H. P. A. Gustafsso n. Iwahori-metaplectic duality. J. Lond. Math. Soc. (2) , 109(6):Paper No. e12896, 54, 2024

    work page 2024

  12. [12]

    Brubaker, D

    B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and P. E. G unnells. Metaplectic ice. In Multiple Dirichlet series, L- functions and automorphic forms , volume 300 of Progr. Math., pages 65–92. Birkh¨ auser/Springer, New York, 2012

    work page 2012

  13. [13]

    Brubaker, D

    B. Brubaker, D. Bump, and S. Friedberg. Schur polynomia ls and the Yang-Baxter Equation. Communications in Mathe- matical Physics , 2011

    work page 2011

  14. [14]

    Brubaker, C

    B. Brubaker, C. Frechette, A. Hardt, E. Tibor, and K. W eb er. Frozen pipes: lattice models for Grothendieck polynomi als. Algebr. Comb. , 6(3):789–833, 2023

    work page 2023

  15. [15]

    Buciumas and T

    V. Buciumas and T. Scrimshaw. Double Grothendieck poly nomials and colored lattice models. Int. Math. Res. Not. IMRN , (10):7231–7258, 2022

    work page 2022

  16. [16]

    Buciumas, T

    V. Buciumas, T. Scrimshaw, and K. W eber. Colored five-ve rtex models and Lascoux polynomials and atoms. J. Lond. Math. Soc. (2) , 102(3):1047–1066, 2020

    work page 2020

  17. [17]

    Bump and S

    D. Bump and S. Naprienko. Colored Bosonic models and mat rix coefficients. Commun. Number Theory Phys. , 18(2):441– 484, 2024

    work page 2024

  18. [18]

    Chari and A

    V. Chari and A. Pressley. A Guide to Quantum Groups . Cambridge University Press, Cambridge, 1994

    work page 1994

  19. [19]

    Chen and Z

    Y. Chen and Z. Zhang. A weak version of Kirillov’s conjec ture on Hecke–Grothendieck polynomials. J. Combin. Theory Ser. A , 186:Paper No. 105555, 18, 2022. 30

    work page 2022

  20. [20]

    M. J. Curran, C. Frechette, C. Yost-W olff, S. W. Zhang, an d V. Zhang. A lattice model for super LLT polynomials. Comb. Theory, 3(2):Paper No. 3, 52, 2023

    work page 2023

  21. [21]

    Di Francesco and P

    P. Di Francesco and P. Zinn-Justin. Inhomogeneous mode l of crossing loops and multidegrees of some algebraic varie ties. Comm. Math. Phys. , 262(2):459–487, 2006

    work page 2006

  22. [22]

    Gorbounov and C

    V. Gorbounov and C. Korff. Quantum integrability and gen eralised quantum Schubert calculus. Adv. Math. , 313:282–356, 2017

    work page 2017

  23. [23]

    A. N. Kirillov. Notes on Schubert, Grothendieck and key polynomials. SIGMA Symmetry Integrability Geom. Methods Appl., 12:Paper No. 034, 56, 2016

    work page 2016

  24. [24]

    Knutson and E

    A. Knutson and E. Miller. Gr¨ obner geometry of Schubert polynomials. Ann. of Math. (2) , 161(3):1245–1318, 2005

    work page 2005

  25. [25]

    Knutson and P

    A. Knutson and P. Zinn-Justin. Schubert puzzles and int egrability i: invariant trilinear forms, 2017, arXiv:1706 .10019

    work page 2017

  26. [26]

    Knutson and P

    A. Knutson and P. Zinn-Justin. Schubert puzzles and int egrability ii: multiplying motivic segre classes, 2021, arXiv:2102.00563

    work page arXiv 2021

  27. [27]

    T. Kojima. Diagonalization of transfer matrix of super symmetry Uq( ˆsl(M + 1|N + 1)) chain with a boundary. Journal of Mathematical Physics , 54(4), Apr 2013

    work page 2013

  28. [28]

    A. Lascoux. The 6 vertex model and Schubert polynomials . SIGMA Symmetry Integrability Geom. Methods Appl. , 3:Paper 029, 12, 2007

    work page 2007

  29. [29]

    L. C. Mihalcea and C. Su. Whittaker functions from motiv ic Chern classes. Transform. Groups, 27(3):1045–1067, 2022. With an appendix by Mihalcea, Su and Dave Anderson

    work page 2022

  30. [30]

    Motegi and K

    K. Motegi and K. Sakai. Vertex models, TASEP and Grothen dieck polynomials. J. Phys. A , 46(35):355201, 26, 2013

    work page 2013

  31. [31]

    J. H. H. Perk and C. L. Schultz. New families of commuting transfer matrices in q-state vertex models. Phys. Lett. A , 84(8):407–410, 1981

    work page 1981

  32. [32]

    Reshetikhin

    N. Reshetikhin. Multiparameter quantum groups and twi sted quasitriangular Hopf algebras. Lett. Math. Phys. , 20(4):331– 335, 1990

    work page 1990

  33. [33]

    A. Sudbery. Consistent multiparameter quantisation o f GL( n). J. Phys. A , 23(15):L697–L704, 1990

    work page 1990

  34. [34]

    Takeuchi

    M. Takeuchi. A two-parameter quantization of GL( n) (summary). Proc. Japan Acad. Ser. A Math. Sci. , 66(5):112–114, 1990

    work page 1990

  35. [35]

    Wheeler and P

    M. Wheeler and P. Zinn-Justin. Littlewood-Richardson coefficients for Grothendieck polynomials from integrabili ty. J. Reine Angew. Math. , 757:159–195, 2019

    work page 2019

  36. [36]

    S. Zemel. Polynomial divided difference operators sati sfying the braid relations, 2024, arXiv:2404.19395. 31

    work page arXiv 2024