Formulas for Koornwinder polynomials

Arun Ram; Laura Colmenarejo; Lucas Gagnon

REVIEW 3 major objections 4 minor 28 references

Formulas for Koornwinder polynomials

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper proves every relative Koornwinder polynomial equals a normalization factor times a weighted sum over compressed set-valued tableaux.

desk verdict A genuinely new CSV-tableau formula for Koornwinder polynomials, worth refereeing, but the proof of Theorem 5.2 rests on an unjustified negative-power identity and a few unfinished cases. read the letter →

arxiv 2608.02810 v1 pith:ODGBUYNU submitted 2026-08-03 math.CO math.RT

Laura Colmenarejo , Lucas Gagnon , Arun Ram This is my paper · ORCID

classification math.COmath.RT MSC 05E0533D52

keywords KoornwinderpolynomialsMacdonalddoubleaffineHeckealgebraset-valuedtableauxalcovewalkscreationformulacompressionsignedpermutations

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 gives explicit monomial expansions of all Koornwinder polynomials—the Macdonald polynomials for the affine root system of type CC_n—including the relative versions indexed by signed permutations. The headline result, Theorem 5.2, expresses each relative Koornwinder polynomial as a normalization factor times a weighted sum over compressed set-valued (CSV) tableaux, with weights built from coroot data, c-functions, and fold functions. Along the way, the paper provides a creation formula in divided-difference operators and a set-valued tableaux reformulation of the alcove walk formula, matching the three classical formulas for type GL_n Macdonald polynomials. If correct, this gives a complete, directly computable combinatorial description of a polynomial family that specializes to Askey-Wilson polynomials and other classical-type orthogonal polynomials.

What carries the argument

The double affine Hecke algebra of type CC_n, represented on Laurent polynomials by operators T_0,...,T_n and X_1,...,X_n, supplies the recursive structure. Creation operators τ_i = T_i + F^+_{α_i}, expressed through c-functions and fold functions, generate Koornwinder polynomials from 1. A box-greedy reduced word u^□_µ for the affine Weyl group element u_µ organizes the diagram of µ into boxes and produces a coroot sequence; the two types of compression sections—around-the-end and across-the-0-gap—then consolidate many alcove-walk choices into fewer CSV-tableaux terms, turning 2^k choices into k+1 or k^2 terms.

What would settle it

Compute both sides of Theorem 5.2 for n = 2, z = 1, µ = (2,1) as Laurent polynomials in x_1, x_2 over the generic parameter field; any disagreement in a single coefficient of a monomial would disprove the formula. Equivalently, verify the braid relation T_0T_1T_0T_1 = T_1T_0T_1T_0 on the polynomial representation; if it fails, the representation is not faithful.

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Extended reading notes

Core claim

Theorem 5.2 states that for every signed permutation z and every composition µ, E^z_µ = nf(z, µ) times the sum of cwt(T) over all compressed set-valued tableaux T of shape µ. The CSV tableaux are set-valued tableaux whose entries satisfy a left-justified rule within each compression section. The compressed weight cwt(T) is a product of section weights, each obtained by evaluating a rational function in Y-variables that is assembled from c-functions and fold functions attached to the coroot sequence of a specially chosen box-greedy reduced word. This completes a Koornwinder analogue of the creation, alcove walk, and non-attacking fillings formulas for type GL_n Macdonald polynomials, in full

Load-bearing premise

The load-bearing premise is that the polynomial action of the double affine Hecke algebra of type CC_n is faithful for generic parameters, so that the operator identities used in the creation formula and in the inductive proof of Theorem 5.2 are legitimate.

Editorial extensions

If this is right

Every relative Koornwinder polynomial E^z_µ, and therefore every symmetric Koornwinder polynomial P_λ by summing E^w_λ over w, has an explicit finite expansion indexed by CSV tableaux.
The expansion specializes to the type GL_n non-attacking fillings formula when the extra parameters collapse, and to set-valued tableaux in the uncompressed limit.
For n = 1 the formula yields monomial expansions of Askey-Wilson polynomials with the standard parameter correspondence.
The creation formula in divided-difference operators gives a direct recurrence that can be implemented computationally, and the paper includes code for the combinatorial constructions.

Reading between the lines

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

The two-compression scheme may extend to other DAHA families, potentially yielding compressed tableaux formulas for type B/C/D Macdonald and Hall–Littlewood polynomials; the paper hints at Hall–Littlewood but does not develop this.
The compressed weights are rational functions rather than simple monomials, so testing coefficientwise positivity or integrality would be a natural next step that the paper does not claim.
The 0-gap compression is closely related to queue-tableaux constructions for open-boundary ASEP; extending the appendix calculation to general µ might produce a bijection between CSV tableaux and rhombic staircase tableaux, a connection the paper leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read this one if you care about Koornwinder or Macdonald expansions. The genuinely new content is the compressed set-valued tableau formula (Theorem 5.2), the two compression mechanisms that produce it, and the set-valued tableau reformulation of the alcove walk for type CC_n. That is a real step beyond the GL_n story, and the paper also gives a clean creation formula via divided differences and a useful box-greedy reduced word. The DAHA appendix is a real service. The paper ships sage code and works several examples by hand, which is more than most submissions in this area do.

The soft spots are real but, I think, repairable. The proof of Theorem 5.2 uses the identity

bE^{z d_{k,i} u(k)}_nu = t^{-(1/2) ell(u(k))} bE^{z d_{k,i}}_nu

and attributes it to (4.5). But (4.5), as written, gives positive powers and assumes length additivity with a right factor stabilizing the index. Here u(k) is not shown to satisfy that, and in the compression cases the relevant length can decrease. This negative-power identity is load-bearing for the t-exponents in cwt(T). I could not find a lemma in Section 5 that proves it. That is a genuine gap in the written proof, not a manufactured one. Given the worked examples and the code, I would bet the formula is correct, but Theorem 5.2 is not fully established as printed.

Proposition 5.10, which handles the general around-the-end compression weights, leaves several cases to "similar" arguments. That would be a minor complaint on its own, but it is exactly where the missing exponent identity would land, so the two issues compound. Appendix B has an apparent typo: in Proposition B.1, the definition of a_m writes both branches as y(m)<0; the second should surely be y(m)>0. Minor, but confusing.

Bottom line: this paper deserves a serious referee. The central claim is plausible and important, the exposition is honest, and the gaps are addressable. I would send it out with a clear request to justify the negative-power step, complete the omitted cases of Proposition 5.10, and fix the Appendix B typo before publication.

Referee Report

3 major / 4 minor

Summary. The paper develops explicit monomial expansions for relative Koornwinder polynomials E^z_μ. It states a creation formula in terms of divided-difference operators (Proposition 4.2), an uncompressed set-valued tableaux formula obtained from the alcove walk method (Theorem 4.6), and, as the main result, a compressed set-valued tableaux formula (Theorem 5.2). The proof is built on the double affine Hecke algebra framework for type CC_n, a new box-greedy reduced word for the element u_μ, c-function and fold-function identities, and two compression mechanisms: across-the-0-gap and around-the-end compression. Appendix B gives a further extension used to match the CMW rhombic staircase tableaux example.

Significance. If Theorem 5.2 is correct, this is a substantial contribution: it provides the first complete compressed tableaux expansion for all relative Koornwinder polynomials, generalizing the type GL_n non-attacking fillings and set-valued tableaux formulas. The paper is also useful for its detailed DAHA exposition, the box-greedy reduced word for type CC_n, and the included Sage code implementing the combinatorial constructions. I see no circularity: the cited c-function identities from [CR25] are parameter-free algebraic lemmas, not equivalent to the new tableaux formula. However, several load-bearing proof obligations are left incomplete, so the central claim is not yet fully established as written.

major comments (3)

[§5.2, proof of Theorem 5.2] The displayed identity bE_{z d_{k,i}u(k)}^ν = t^{-(1/2)ℓ(u(k))} bE_{z d_{k,i}}^ν is attributed to (4.5), but (4.5) gives positive powers only for a length-additive decomposition with a right factor that stabilizes the index. To obtain the negative-power identity one must prove either that u(k) stabilizes ν and that ℓ(z d_{k,i}u(k)) = ℓ(z d_{k,i}) + ℓ(u(k)), or apply (4.5) to u(k)^{-1}. No such verification is given. Moreover, the derivation in this passage writes the (4.5) normalization as a single t^{(1/2)ℓ} power, although (4.5) has separate t^{(1/2)ℓ_s} and t_n^{(1/2)ℓ_d} factors, while the weights in (5.5)–(5.6) are defined using ℓ_s and explicit t_n factors. This bookkeeping is load-bearing because every coefficient cwt(T) in the CSV expansion depends on these t- and t_n-powers.
[§5.5, Proposition 5.10] The proof of Proposition 5.10 treats Case 1 and Case 2 in detail and then says that the remaining cases are similar. Cases 3 and 4 correspond to j = -n and j = -m and require the boundary generator s_n, the recurrences (5.16)–(5.17), and the t_n-dependent terms in (2.24)–(2.25). Since Proposition 5.10 is one of the two compression lemmas on which Theorem 5.2 is built, the omitted cases are not merely cosmetic. The proof should be completed, or at least reduced to the displayed Case 2 calculations with the boundary changes made explicit.
[§5.4–§5.5, weights for around-the-end compression] The definition of covid(z_S,k) in (5.5)–(5.6) uses ℓ_s(v) - ℓ_s(u(k)), but the proof of Theorem 5.2 uses ℓ(v) - ℓ(u(k)) before switching to covid. For around-the-end compression, v can involve s_n, so ℓ_d does not automatically vanish. The additional t_n-factors appearing in (5.5) appear to be intended to absorb this, but the connection is not demonstrated. A precise accounting of ℓ_s and ℓ_d in the induction step is needed to make Theorem 5.2 follow from the stated lemmas.

minor comments (4)

[§2.6, Eq. (2.20)] In the definition of A^{(i)}_i(β), the exponent n-s-1 uses an undefined summation index s; from the surrounding formulas it should presumably be n-w-1.
[§2.6, (CS2)] The line 'Let i, m ∈ {1, . . . , m} with i < m' should be 'Let i, m ∈ {1, . . . , n} with i < m'.
[Appendix B, Proposition B.1] In the definition of a_m, the two cases are both written as 'ify(m)<0'; the second should be 'ify(m)>0'.
[§5.4.2, proof of Proposition 5.7] The proof uses color-coded terms as a visual aid; please ensure the exposition remains readable in black and white, since the distinction between the colored families is essential for following the calculation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 5.2 is derived by induction from the DAHA creation/alcove-walk steps, not assumed from a fit or from a self-citation chain.

full rationale

The derivation chain is self-contained. Theorem 5.2 is proved by induction on compression sections, and the inductive steps are Propositions 5.5 and 5.10; these are proved from Proposition 4.3, which in turn is proved directly from the creation operators and the DAHA facts in Proposition A.7. The only external inputs are the polynomial representation of the DAHA (equations (1.1)-(1.3)), the c-function identities (2.16)-(2.17), and the standard creation formula. These are parameter-free algebraic facts; [CR25] and [Mac03] are cited for them, and [GR21] is cited for the compression heuristic, but the paper does not import the target CSV expansion from any citation. No parameter is fitted to data and then renamed a prediction: the normalization factor nf(z,mu) is fixed by (1.8) from the coefficient-of-x^{z mu} condition, not by the tableaux sum. The weights cwt(T) in (5.5)-(5.7) are defined as packaged expressions and then proven, in Propositions 5.5, 5.7, and 5.10, to equal the coefficients obtained from the DAHA recursion; the proof does not assume that equality. The possible gap noted in the use of (4.5) with a negative exponent is a proof-completeness concern, not a circularity: (4.5) is a consequence of stabilizer normalization and is not the target formula. No step reduces to a self-citation that is itself unverified or to a parameter fitted on the target quantity.

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

The paper introduces no fitted parameters or speculative entities. The Koornwinder parameters q,t,t0,u0,tn,un are fixed inputs from the standard theory. The new combinatorial objects (CSV-tableaux, compression sections) are definitions within the proof, not independent postulates.

assumptions (3)

domain assumption The operators T_0,...,T_n and X_1,...,X_n defined in (1.1)-(1.3) satisfy all relations of the double affine Hecke algebra of type CC_n, and the polynomial representation is faithful for generic parameters.
Invoked in Section 1.1 after (1.3); underpins the intertwiners, the hexagon basis lemma, and all recursive steps leading to Theorem 5.2.
domain assumption The evaluation homomorphism ev_mu defined in (2.11) is well-defined for generic values of the parameters, so denominators in c-functions and fold functions do not vanish.
Used throughout Sections 4 and 5 to turn Y-operator eigenvalues into scalar coefficients in the tableaux weights.
standard math Known results from the DAHA theory of Koornwinder polynomials: the creation formula (Macdonald), the alcove walk formula (Ram-Yip), and the c-function identities of [CR25].
These are cited inputs, e.g., Theorem 4.6 reparametrizes the Ram-Yip alcove walk, and Lemma 5.9 uses [CR25] identities. They are independent of the new compressed formula.

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Cite this review

Pith. "Pith review of Formulas for Koornwinder polynomials." pith.science (2026). https://pith.science/paper/ODGBUYNU

@misc{pith2026260802810,
  author       = {Pith},
  title        = {Pith review of: Formulas for Koornwinder polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ODGBUYNU}},
  note         = {Machine review of arXiv:2608.02810}
}

abstract

This paper provides formulas for Koornwinder polynomials in analogy with the creation formula, the alcove walk formula and the non-attacking fillings formula for the type $GL_n$ Macdonald polynomials. We state the creation formula in terms of the divided-difference operators used in Schubert calculus, and we use a box-greedy reduced word to reformulate the alcove walk formula in terms of uncompressed set-valued tableaux. Then two types of compression, ``around-the-end compression'' and ``across-the-$0$-gap compression'', are used to derive a formula for Koornwinder polynomials in terms of compressed set-valued tableaux. Throughout we work in the full generality of relative Koornwinder polynomials, which are the analogues of the permuted basement Macdonald polynomials used in the type $GL_n$ case.

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Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Non-symmetric Macdonald polynomials and Demazure-Lusztig operators

P.\ Alexandersson, Non-symmetric Macdonald polynomials and Demazure-Lusztig operators, S\'em.\ Lothar.\ Combin. B76d (2019), MR4063906, arXiv:1602.05153

work page Pith review arXiv 2019

[2] [2]

Bourbaki, Groupes et alg\`ebres de Lie , vol

N. Bourbaki, Groupes et alg\`ebres de Lie , vol. 4--6, Masson 1981, MR0647314

work page 1981

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L.\ Cantini, J.\ de Gier and M.\ Wheeler, Matrix product formula for Macdonald polynomials, J.\ Phys.\ A Math.\ Theor., 48 (2015) 384001, arXiv:1505.00287

work page Pith review arXiv 2015

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L.\ Cantini, A.\ Garbali, J.\ de Gier and M.\ Wheeler, Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries, J.\ Phys.\ A49(2016), no.44, 444002, 23 pp., MR3562836, arXiv: 1607.00039

work page Pith review arXiv 2016

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Colmenarejo, L

L. Colmenarejo, L. Gagnon and A. Ram, ``KoornwinderPolynomials,'' GitHub repository, 2026. [Online]. Available: https://github.com/lucasgagnon/KoornwinderPolynomials

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