Okounkov (Adv. Appl. Math. 1998) classified the perfect grids of symmetric Newton interpolation: among grids whose knot coordinates depend only on the corresponding parts of the indexing partition, the extra vanishing property holds exactly for the factorial monomial, factorial Schur, and (BC-type) interpolation Macdonald grids and their degenerations, so any other exactly solvable grid would require an entirely new type of formula. The paper explicitly suggested extending this characterization to the nonsymmetric Macdonald interpolation polynomials. The extension remains open. In type A, the nonsymmetric interpolation Macdonald polynomials, their extra vanishing with respect to Knop's partial order on compositions, binomial formulas, and a duality theorem were established by Knop, Sahi, Lascoux–Rains–Warnaar, and Sahi–Stokman, but no grid-level classification exists. In the BC (Laurent-polynomial) setting, Disveld–Koornwinder–Stokman constructed nonsymmetric interpolation Laurent polynomials recovering Okounkov's BC-type polynomials under Hecke symmetrization, but even their extra vanishing property is only a computer-supported conjecture, and no published proof or disproof is known. On the symmetric side, Korotkikh (Selecta Math. 2024) discovered genuinely new perfect grids of a difference form, realized by spin q-Whittaker interpolation polynomials with vertex-model formulas, classified them for rank at least 3, and left rank 2 open. The problem asks for the nonsymmetric perfect-grid characterization, together with a proof or disproof of the Disveld–Koornwinder–Stokman extra-vanishing conjecture as the concrete open core of its BC branch; the loosely specified extra-triangularity classification raised in the source's footnote is retained as a secondary component.

Background

Macdonald polynomials are a distinguished two-parameter (q,t)(q,t) basis of the algebra of symmetric polynomials, indexed by partitions λ=(λ1≥⋯≥λn≥0)\lambda=(\lambda_1\ge\dots\ge\lambda_n\ge0); they simultaneously generalize Schur, Hall–Littlewood, and Jack polynomials and are tightly connected to the representation theory of double affine Hecke algebras (q-deformations of the algebra of affine Weyl group symmetries). In the mid-1990s, Knop and Sahi introduced inhomogeneous interpolation (also called shifted) analogues: symmetric polynomials whose top homogeneous parts are the Macdonald polynomials and which are characterized by Newton-type interpolation conditions rather than by orthogonality. The source paper (Okounkov 1998) sets up the following general framework. Over a field kk, a grid is a function Ω:{1,…,n}×Z≥0→k\Omega:\{1,\dots,n\}\times\mathbb{Z}_{\ge0}\to k, and to each partition λ\lambda of length at most nn is attached the knot λˉ=(Ω(1,λ1),…,Ω(n,λn))∈kn\bar\lambda=(\Omega(1,\lambda_1),\dots,\Omega(n,\lambda_n))\in k^n (the source itself indexes a grid on {0,…,n}\{0,\dots,n\} with partitions of length at most n+1n+1; we use the one-based convention of the later literature). The Newton interpolation polynomial Pλ(x;Ω)P_\lambda(x;\Omega) is the essentially unique symmetric polynomial of degree at most ∣λ∣|\lambda| vanishing at all knots μˉ\bar\mu with ∣μ∣≤∣λ∣|\mu|\le|\lambda|, μ≠λ\mu\ne\lambda (non-degeneracy of the grid — a generic pairwise-distinctness condition on the knot coordinates — guarantees existence). Okounkov calls a grid perfect when its polynomials enjoy the extra vanishing property Pλ(μˉ;Ω)=0P_\lambda(\bar\mu;\Omega)=0 unless λ⊆μ\lambda\subseteq\mu (inclusion of Ferrers diagrams) — that is, they vanish for free at knots where the defining conditions do not require it. His Main Theorem lists all perfect grids among those of this product form: the factorial monomial grid [ij]=cj[ij]=c_j, the factorial Schur grid [ij]=cj−i[ij]=c_{j-i}, and the interpolation Macdonald grid [ij]=a+bqjti+cq−jt−i[ij]=a+bq^jt^i+cq^{-j}t^{-i} (with pairwise-distinctness conditions), together with their degenerations. The interpolation Macdonald polynomials here are Okounkov's BC-type family, whose top homogeneous parts are Koornwinder polynomials (the analogues of Macdonald polynomials for the root system of type BC, the signed-permutation symmetry type) and which contains the Knop–Sahi type-A interpolation polynomials as the degeneration [ij]=qjtn−i[ij]=q^jt^{n-i}. Perfect grids are precisely the exactly solvable cases: for them the interpolation polynomials admit a tableau-sum formula (a weighted sum over reverse Young tableaux) and a qq-integral representation (a multiple Jackson-type q-hypergeometric summation), and Okounkov derives from his classification that no grid outside his list admits a tableau-sum formula of that type or an analogous qq-integral representation.

Nonsymmetric polynomials. The nonsymmetric Macdonald polynomials EαE_\alpha, indexed by compositions α∈Z≥0n\alpha\in\mathbb{Z}_{\ge0}^n rather than partitions, were introduced by Cherednik; they diagonalize the Y-operators (a commuting family of q-difference operators) of the double affine Hecke algebra and symmetrize to Macdonald polynomials. Knop and Sahi constructed their interpolation analogues: nonsymmetric polynomials satisfying Newton conditions at composition-indexed knots, with a binomial formula (Sahi) and an extra vanishing property governed by a new partial order on compositions (Knop). The source paper closes its introduction by suggesting that these nonsymmetric polynomials might admit a similar characterization — a classification of the nonsymmetric grids that are perfect. In the BC (Laurent-polynomial) setting, Disveld–Koornwinder–Stokman introduced nonsymmetric interpolation Laurent polynomials Gα(x;q,τ)G_\alpha(x;q,\tau) for the grid τiqm+(τiqm)−1\tau_iq^m+(\tau_iq^m)^{-1}, recovering Okounkov's BC-type interpolation Macdonald polynomials by symmetrization with respect to a type CnC_n Hecke algebra action (the q-analogue of averaging over the Weyl group of signed permutations); their extra vanishing is conjectural: the paper states explicitly that it is not known whether these polynomials satisfy extra vanishing properties, its appendix describes the conjectural zero sets by explicit inequalities on the index lattice Zn\mathbb{Z}^n, and it suggests a hyperoctahedral analogue of Knop's order (the counterpart of his partial order for the group of signed permutations, the Weyl group of types B and C) as the expected order-theoretic formulation, with supporting computer-algebra evidence only in rank two and degree at most 4.

Spin q-Whittaker polynomials and new grids. Inhomogeneous spin q-Whittaker polynomials, constructed as partition functions of integrable vertex models (exactly solvable two-dimensional lattice models whose local weights satisfy the Yang–Baxter equation), generalize the t=0t=0 Macdonald polynomials by two additional parameter sequences. Korotkikh characterized them by vanishing at knots whose ii-th coordinate depends on the difference λi−λi+1\lambda_i-\lambda_{i+1} of consecutive partition parts — a grid form outside Okounkov's product form — constructed interpolation analogues satisfying extra vanishing, and classified all perfect grids of that difference form for n≥3n\ge3, while numerical evidence points to additional perfect grids in rank 2.

The source paper also notes, in a footnote, a broader symmetric question: describe all orthogonal or interpolation symmetric polynomials satisfying an extra triangularity condition. Here extra triangularity means that the output of the orthogonalization (or interpolation) process does not depend on which total order on the index set is chosen, as long as that total order refines the dominance order on partitions (the standard partial order in which λ≥μ\lambda\ge\mu iff λ1+⋯+λi≥μ1+⋯+μi\lambda_1+\dots+\lambda_i\ge\mu_1+\dots+\mu_i for every ii) — a condition strictly stronger than ordinary triangularity with respect to one fixed order, which ordinary Macdonald polynomials famously enjoy.

Finally, the source paper draws a heuristic analogy between extra vanishing and the "extra integrability" of exactly solvable many-body systems — systems possessing more independent integrals of motion than integrability requires — noting that new examples of such systems were found after extra integrability had been conjectured to characterize exact solvability; the analogy suggests that analogous surprises, in the form of further exactly solvable interpolation grids, may likewise surface.

Problem Statement

The common objective is to determine which nonsymmetric Newton interpolation structures — grids and their interpolation polynomial families — possess the extra vanishing property, extending Okounkov's characterization of the exactly solvable symmetric cases in the direction the source paper explicitly suggested. Concretely, the problem comprises the following targets. (1) Nonsymmetric characterization (the question explicitly raised at the end of Okounkov's introduction): formulate the nonsymmetric Newton interpolation scheme on the algebra of all polynomials, respectively Laurent polynomials, with knots indexed by compositions in the polynomial (type-A) setting and by lattice points of Zn\mathbb{Z}^n in the BC (Laurent-polynomial) setting, and classify the nonsymmetric grids that are perfect, in the sense that their interpolation polynomials satisfy the appropriate extra vanishing property. The classification is sought in both the type-A and the BC settings, up to degenerations — the grid limits obtained by specializing the parameters, as in Okounkov's symmetric Main Theorem. (2) Component conjecture: prove or disprove that the nonsymmetric BC-type interpolation Macdonald polynomials Gα(x;q,s,t)G_\alpha(x;q,s,t) (the specialization at τi=stn−i\tau_i=st^{n-i} of the nonsymmetric interpolation Laurent polynomials Gα(x;q,τ)G_\alpha(x;q,\tau) introduced in the Background) satisfy the extra vanishing property conjectured by Disveld–Koornwinder–Stokman — a conjectural description of their zero sets by explicit inequality-defined regions of the index lattice Zn\mathbb{Z}^n — currently supported only by computer experiments in rank two and degree at most 4. (3) Secondary and loosely specified in the source's footnote: describe all orthogonal or interpolation symmetric polynomials satisfying the extra triangularity condition, meaning that the orthogonalization or interpolation output is independent of the choice of total order on the index set as long as it refines the dominance order on partitions. (1)–(2) share the completion boundary of a rigidity classification for nonsymmetric interpolation polynomials with extra vanishing, with (2) the concrete BC component of (1): settling (1) resolves (2), while settling (2) alone, or settling (1) only in the type-A setting, is partial progress on the common objective. (3) is the broader symmetric variant of the same rigidity phenomenon raised in the source's footnote and is not part of that completion boundary.

References

  1. Andrei Okounkov, “On Newton Interpolation of Symmetric Functions: A Characterization of Interpolation Macdonald Polynomials”, Advances in Applied Mathematics 20 (1998), no. 4, 395–428. DOI: 10.1006/aama.1998.0590, https://doi.org/10.1006/aama.1998.0590 (arXiv:q-alg/9712052, https://arxiv.org/abs/q-alg/9712052)
  2. Ivan Cherednik, “Nonsymmetric Macdonald polynomials”, International Mathematics Research Notices 1995, no. 10, 483–515. DOI: 10.1155/S1073792895000341, https://doi.org/10.1155/S1073792895000341
  3. Siddhartha Sahi, “Interpolation, integrality, and a generalization of Macdonald's polynomials”, International Mathematics Research Notices 1996, no. 10, 457–471. DOI: 10.1155/S107379289600030X, https://doi.org/10.1155/S107379289600030X
  4. Friedrich Knop and Siddhartha Sahi, “Difference equations and symmetric polynomials defined by their zeros”, International Mathematics Research Notices 1996, no. 10, 473–486. DOI: 10.1155/S1073792896000311, https://doi.org/10.1155/S1073792896000311 (arXiv:q-alg/9610017, https://arxiv.org/abs/q-alg/9610017)
  5. Friedrich Knop, “Symmetric and non-symmetric quantum Capelli polynomials”, Commentarii Mathematici Helvetici 72 (1997), no. 1, 84–100. DOI: 10.4171/cmh/72.1.7, https://doi.org/10.4171/cmh/72.1.7
  6. Andrei Okounkov, “BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials”, Transformation Groups 3 (1998), no. 2, 181–207. DOI: 10.1007/BF01236432, https://doi.org/10.1007/BF01236432 (arXiv:q-alg/9611011, https://arxiv.org/abs/q-alg/9611011)
  7. Siddhartha Sahi, “The binomial formula for nonsymmetric Macdonald polynomials”, Duke Mathematical Journal 94 (1998), no. 3, 465–477. DOI: 10.1215/s0012-7094-98-09419-4, https://doi.org/10.1215/s0012-7094-98-09419-4
  8. Alain Lascoux, Eric M. Rains, and S. Ole Warnaar, “Nonsymmetric interpolation Macdonald polynomials and gl_n basic hypergeometric series”, Transformation Groups 14 (2009), no. 3, 613–647. DOI: 10.1007/s00031-009-9061-1, https://doi.org/10.1007/s00031-009-9061-1 (arXiv:0807.1351, https://arxiv.org/abs/0807.1351)
  9. Siddhartha Sahi and Jasper V. Stokman, “Some Remarks on Non-Symmetric Interpolation Macdonald Polynomials”, International Mathematics Research Notices 2021, no. 19, 14814–14839. DOI: 10.1093/imrn/rnz229, https://doi.org/10.1093/imrn/rnz229 (arXiv:1901.04852, https://arxiv.org/abs/1901.04852)
  10. Niels Disveld, Tom H. Koornwinder, and Jasper V. Stokman, “A nonsymmetric version of Okounkov's BC-type interpolation Macdonald polynomials”, Transformation Groups 26 (2021), no. 4, 1261–1292. DOI: 10.1007/s00031-021-09672-x, https://doi.org/10.1007/s00031-021-09672-x (arXiv:1808.01221, https://arxiv.org/abs/1808.01221)
  11. Sergei Korotkikh, “Representation theoretic interpretation and interpolation properties of inhomogeneous spin q-Whittaker polynomials”, Selecta Mathematica (New Series) 30 (2024), no. 3, article 40. DOI: 10.1007/s00029-024-00930-w, https://doi.org/10.1007/s00029-024-00930-w (arXiv:2204.06166, https://arxiv.org/abs/2204.06166)
  12. Siddhartha Sahi, Jasper V. Stokman, and Vidya Venkateswaran, “Quasi-polynomial representations of double affine Hecke algebras”, arXiv:2204.13729 (2022), https://arxiv.org/abs/2204.13729
  13. Jasper V. Stokman, “Quasi-Polynomial Extensions of Nonsymmetric Macdonald–Koornwinder Polynomials”, Symmetry, Integrability and Geometry: Methods and Applications 21 (2025), paper 086. DOI: 10.3842/sigma.2025.086, https://doi.org/10.3842/sigma.2025.086
  14. Houcine Ben Dali and Lauren Williams, “A combinatorial formula for Interpolation Macdonald polynomials”, arXiv:2510.02587 (2025), https://arxiv.org/abs/2510.02587
  15. Ajeeth Gunna, Michael Wheeler, and Paul Zinn-Justin, “Inhomogeneous q-Whittaker Polynomials I: Duality and Expansions”, arXiv:2512.04468 (2025), https://arxiv.org/abs/2512.04468
  16. S. V. Kerov, “Generalized Hall–Littlewood symmetric functions and orthogonal polynomials”, in Representation Theory and Dynamical Systems (A. M. Vershik, ed.), Advances in Soviet Mathematics 9, American Mathematical Society, Providence, RI, 1992, pp. 67–94. zbMATH: 0760.05089, https://zbmath.org/?q=an%3A0760.05089 (the nearest antecedent of the extra-triangularity question, cited as [K] in the source paper)