{"id":"c1991adc-c26a-42ba-a15b-2e31a82f8e08","arxiv_id":"2608.02810","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New uncompressed and compressed set-valued tableaux formulas give explicit monomial expansions for all relative Koornwinder polynomials.","lead":"This paper derives new combinatorial formulas, including a compressed set-valued tableaux expansion, for Koornwinder polynomials, the universal classical-type Macdonald polynomials. The new formulas generalize known type GL_n expansions and could simplify computations in integrable probability and orthogonal polynomials.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.2's induction uses a negative t-power normalization attributed to (4.5), which only gives positive powers; this is the coefficient of cwt(T) and is unproved.","rationale":"The reader's stated weakest assumption is the validity/faithfulness of the polynomial representation of the DAHA. That is standard background and is cited to Noumi, Sahi, and others, so I do not treat it as the main load-bearing concern. The more concrete risk is internal to the proof of Theorem 5.2: the induction step invokes (4.5) for a relation with a negative exponent, while (4.5) as stated gives positive exponents under length-additivity assumptions. The compressed weights in (5.5)-(5.6) are exactly these powers, so an unproved normalization identity would invalidate the central formula even if the DAHA representation is correct. The paper may be right; indeed the worked Example 5.3 is consistent with the negative-power convention. But the proof as written lacks the needed lemma, and the terse 'remaining cases are similar' in Proposition 5.10 does not supply it. I therefore do not recommend changing the reader's conditional verdict; the paper should be published only after the missing normalization lemma and the case checks are supplied.","tokens_in":74986,"tokens_out":26824,"duration_ms":271285,"concrete_test":"Check the disputed normalization in the smallest around-the-end cases. For n=2, i=1, mu=(-1,0), nu=(1,0), compute E^z_mu by the creation formula (4.4)/(E1)-(E3) and by the Theorem 5.2 CSV sum, for z one-line entries (-1,2) (case j=-i) and (-2,1) (case j=-n). Compare the coefficients as rational functions in q, t, t_0, t_n, u_0, u_n. If the two computations agree, the negative-power identity is true in these cases but still needs a proof; if they disagree, Theorem 5.2 is false. The same check can be run for all z in W_fin and |mu_i|<=2 using the authors' Sage code.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 5.2: E^z_mu is a sum over CSV-tableaux with weights cwt(T). Those weights are built from factors t^{1/2 covid(z,k)} in (5.5)-(5.6), so the exact t- and t_n-powers in the induction step are load-bearing. In the proof of Theorem 5.2, after setting z = yv and u(k) = d_{k,i}^{-1} v^{-1} d_{ell,i}, the text asserts\n\n  bE^{z d_{k,i} u(k)}_nu = t^{-(1/2) ell(u(k))} bE^{z d_{k,i}}_nu\n\nand attributes this to (4.5). But (4.5) gives positive powers t^{+(1/2) ell(v)} only for decompositions with length additivity and a right factor stabilizing the index. Here u(k) is not shown to satisfy length additivity with z d_{k,i}; in compression sections the relevant length typically decreases. Thus the displayed negative-power identity is a separate assertion, not a special case of (4.5). No lemma in Section 5 justifies it. Since any sign or exponent error here propagates directly to every coefficient in the CSV expansion, the formula is not fully established as written. The later case analysis in Proposition 5.10, whose proof leaves several cases to 'similar' arguments, does not repair this gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":75341,"tokens_out":8620,"duration_ms":97340,"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":[{"comment":"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.","section":"§5.2, proof of Theorem 5.2"},{"comment":"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.","section":"§5.5, Proposition 5.10"},{"comment":"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.","section":"§5.4–§5.5, weights for around-the-end compression"}],"minor_comments":[{"comment":"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.","section":"§2.6, Eq. (2.20)"},{"comment":"The line 'Let i, m ∈ {1, . . . , m} with i < m' should be 'Let i, m ∈ {1, . . . , n} with i < m'.","section":"§2.6, (CS2)"},{"comment":"In the definition of a_m, the two cases are both written as 'ify(m)<0'; the second should be 'ify(m)>0'.","section":"Appendix B, Proposition B.1"},{"comment":"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.","section":"§5.4.2, proof of Proposition 5.7"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in spirit and the main formula is an important target, but the proof of Theorem 5.2 contains a real gap in the use of (4.5), and Proposition 5.10 leaves several boundary cases unproved. These are fixable within the manuscript's scope, so I recommend major revision rather than rejection. I do not see a circularity problem in the use of [CR25]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe soft spots are real but, I think, repairable. The proof of Theorem 5.2 uses the identity\n\nbE^{z d_{k,i} u(k)}_nu = t^{-(1/2) ell(u(k))} bE^{z d_{k,i}}_nu\n\nand 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.\n\nProposition 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.\n\nBottom 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.","headline":"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.","tokens_in":642,"tokens_out":1640,"would_cite":true,"duration_ms":44667,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","33D52"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves every relative Koornwinder polynomial equals a normalization factor times a weighted sum over compressed set-valued tableaux.","keywords":["Koornwinder polynomials","Macdonald polynomials","double affine Hecke algebra","set-valued tableaux","alcove walks","creation formula","compression","signed permutations"],"falsifier":"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.","tokens_in":74922,"feed_emoji":"🧮","tokens_out":4772,"duration_ms":49825,"temperature":0.7,"pith_summary":"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.","feed_headline":"All Koornwinder polynomials fit a tableaux formula","feed_subtitle":"New expansion over compressed set-valued tableaux covers every relative case and the symmetric ones too.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the double affine Hecke algebra framework, creation operators, and orthogonal polynomial background that the paper's recursive method is built on.","marker":"[Mac03]"},{"why":"Provides the box-greedy reduced word, the compression method, and the type GL_n analogues that the Koornwinder formulas generalize.","marker":"[GR21]"},{"why":"Gives the alcove walk formula for Macdonald polynomials that Theorem 4.6 reparametrizes in terms of set-valued tableaux.","marker":"[RY08]"},{"why":"Introduces the c-functions and fold functions used to define the compressed section weights and the electronic Koornwinder polynomials.","marker":"[CR25]"},{"why":"Gives the set-valued tableaux formula for type GL_n Macdonald polynomials that the uncompressed tableaux theorem generalizes.","marker":"[DR22]"},{"why":"Supplies the rhombic staircase tableaux and open-boundary ASEP context, and is extended by the 0-gap compression in Appendix B.","marker":"[CMW23]"},{"why":"Gives the explicit operator form of the Koornwinder DAHA that appears in the definitions of T_0,...,T_n.","marker":"[Nou95]"}],"fun_headline_variants":["Koornwinder polynomials: one tableaux formula for all","CSV tableaux yield Koornwinder formulas","Complete Koornwinder formula via compressed tableaux","Koornwinder analogue of Macdonald tableau formulas","Expansion for every relative Koornwinder polynomial"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Koornwinder polynomials: one tableaux formula for all","CSV tableaux yield Koornwinder formulas","Complete Koornwinder formula via compressed tableaux","Koornwinder analogue of Macdonald tableau formulas","Expansion for every relative Koornwinder polynomial"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000233,"raw_usage":{"total_tokens":1294,"prompt_tokens":671,"completion_tokens":623,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":415,"completion_tokens_details":{"reasoning_tokens":546}},"tokens_in":415,"tokens_out":623,"duration_ms":5885,"temperature":1.0,"reasoning_tokens":546,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T00:46:53.064646+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}