{"id":"9743f6c6-cdaa-4d18-ae36-6128995f2e7b","arxiv_id":"2505.14626","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For k=0,1,2 the equivariant Chern character operators on Hilbert schemes of C^2 match vertex-operator coefficients, giving a partial verification of Okounkov's conjecture and new leading-term formulas.","lead":"This paper proves that, for the equivariant cohomology of Hilbert schemes of points in the affine plane, the first three Chern character operators coincide with coefficients of a deformed vertex operator built from Macdonald and Jack symmetric functions. It uses this identification to partially verify Okounkov's conjecture that certain reduced generating series are q-analogues of multiple zeta values.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.11's qMZV conclusion is delegated to the unpublished [Qin3]; if its bracket statements are inapplicable to (5.24)/(5.30), the paper's partial verification of Okounkov's conjecture does not follow.","rationale":"The reader's weakest_assumption identifies exactly the most load-bearing gap: the partial verification of Okounkov's conjecture in Theorem 5.11 passes through [Qin3], an unpublished preprint by one of the authors, for the decisive qMZV and qBD membership statements. I checked the surrounding argument in good faith. The combinatorial identity in Theorem 4.5 (δ(λ̃)+ℓ(μ̃)=ℓ(μ) for nonzero coefficients) looks suspect at first, but it is valid: for a nonzero coefficient, every positive-part annihilation operator in λ̃ must find a matching creation operator in μ̃, so the number of annihilations equals ℓ(λ̃_+), giving the claimed length relation. Lemma 5.10's reduction at m=0 is also plausible through the diagonal part of W(L,z) and the vanishing of off-diagonal contributions against the diagonal Chern character operators. Thus the internal vertex-operator theorem is not the main risk. The real soft spot is the external black box: the proof of Theorem 5.9 and Theorem 5.11 supplies exact expressions but does not prove the bracket-membership or qMZV expansion statements, citing [Qin3] instead. Since these statements are the final and only bridge from the vertex-operator formalism to Okounkov's conjecture, the concern is load-bearing. A conditional verdict is appropriate: the paper's structural logic is clear, but the decisive external dependency must be verified or supplied before the partial verification can be regarded as complete. I would not reject, because nothing in the presented argument contradicts the cited results, and the N=1 case is likely tractable to check directly.","tokens_in":33381,"tokens_out":19278,"duration_ms":175284,"concrete_test":"Obtain the [Qin3] preprint, or independently test the invoked statement: for N=1 and k=2,3,4, compute the right-hand side of (5.30) directly by expanding the q-products and performing the finite coefficient extraction, then verify explicitly that every coefficient in t1 and t2 is a qMZV of the claimed weight. Additionally, re-derive the product expansion in Theorem 5.11(ii) for 1 - (q)_∞(q̃ t̃^{-1} q)_∞ / ((q̃ q)_∞ (t̃^{-1} q)_∞) to order z^{k+1}, confirming that the coefficients h_m(α) lie in qMZV[α] with leading term [m+2]. If these direct checks pass, the [Qin3] dependency is benign; if they fail, Theorem 5.11(ii) is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.9 and Theorem 5.11 reduce the problem to the explicit coefficient extractions (5.24) and (5.30), then invoke [Qin3, Theorem 1.1, Theorem 1.3(i)-(ii), Remark 3.2] for the decisive conclusions: membership in BD(t1,t2)[m] and qBD[t1,t2], the qMZV expansion with weight bounds, and the leading coefficient (-1)^k [k+2]. The present paper contains no proof of these invoked results, and [Qin3] is not available for independent inspection. If [Qin3]'s hypotheses are not satisfied by the particular coefficient extractions in (5.24)/(5.30) - for instance, if the pole orders in α, the number of z-variables, or the required weight bounds differ from those assumed there - then Theorem 1.4(i)-(ii) and Theorem 5.11(i)-(ii) are unsupported. This is the load-bearing step for the claimed partial verification of Okounkov's conjecture; by contrast, the vertex-operator identification in Theorem 1.2 is largely self-contained once Lemmas 4.2-4.4 are accepted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies equivariant Chern character operators G_k(t_1,t_2) acting on the equivariant cohomology H'_X of the Hilbert schemes (C^2)^[n]. Its main structural result, Theorem 1.2, identifies G_k for 0 <= k <= 2 with the vertex-operator expression t_2^k t_1^{delta(.)} B_k^{(alpha)} evaluated at alpha = -t_1/t_2. The proof uses Jack/Macdonald eigenfunctions and the deformed vertex operators of Cheng--Wang. The paper then applies this identification to Okounkov's conjecture on reduced generating series of equivariant intersections, proving Theorem 1.4, a partial verification whose decisive qBD/qMZV membership steps are delegated to the unpublished preprint [Qin3]. A further section derives closed formulas for higher derivatives of equivariant Heisenberg operators, including the leading-term formula in Proposition 6.5.","tokens_in":33662,"tokens_out":12991,"duration_ms":121561,"significance":"If Theorem 1.2 stands, it is a valuable and concrete structural result: it gives explicit formulas (1.13), (1.14), and (4.40), with no fitted parameters, and it connects the first three equivariant Chern character operators to known vertex-operator constructions. The derivative formula in Theorem 6.4 and the leading-term statement in Proposition 6.5 also appear to be new and are derived self-containedly from Theorem 1.2. The Okounkov verification is explicitly partial, and the paper is honest about that; however, the advertised qBD/qMZV conclusions currently rest on the quoted results of [Qin3], which is not available for inspection. The theorems that are proved in the manuscript are presented with detailed computations, and I found no free parameters or circular definitions in the vertex-operator identification itself.","major_comments":[{"comment":"The central verification of Okounkov's conjecture is not proved in this manuscript. After reducing the problem to the explicit coefficient extractions (5.24) and (5.30), the proofs of [ch_{k_1}...ch_{k_N}]'_{m,t_1,t_2} in BD(t_1,t_2)[m], of <ch_{k_1}...ch_{k_N}>'_{0,t_1,t_2} in qBD[t_1,t_2], and of the weight bounds are all delegated by one-line citations to [Qin3, Theorem 1.1, Theorem 1.3(i)-(ii), Remark 3.2]. [Qin3] is an unpublished preprint by one of the authors, and its statements and hypotheses are not reproduced. As it stands, Theorem 1.4 is conditional on external results whose applicability to (5.24) and (5.30) cannot be checked by the reader. This is a load-bearing dependency: without [Qin3], the paper's partial verification of Okounkov's conjecture does not follow from the presented arguments. The authors should either include full statements and proofs of the invoked results or reformulate the theorems as conditional statements.","section":"Section 5, Theorem 5.9 and Theorem 5.11"},{"comment":"The proof of the qMZV statement for <ch_k>'_{0,t_1,t_2} relies on the expansion 1 - (q)_infinity(\\tilde q \\tilde t^{-1} q)_infinity / ((\\tilde q q)_infinity(\\tilde t^{-1} q)_infinity) = -alpha z^2 \\sum_{m>=0} h_m(alpha) z^m, together with the asserted properties that h_m(alpha) in qMZV[alpha], deg_alpha h_m = m, weight(h_m)=m+2, and the coefficient of alpha^m in h_m equals [m+2]. These properties are quoted from [Qin3, Remark 3.2] with no derivation. This expansion is exactly what converts the vertex-operator coefficient extraction into the qMZV membership and the stated leading coefficient (-1)^k[k+2], so it is load-bearing for Theorem 1.4(ii). The manuscript should supply a proof of this expansion and of the stated properties of h_m, or state the exact hypotheses under which [Qin3, Remark 3.2] applies.","section":"Section 5, Theorem 5.11(ii)"}],"minor_comments":[{"comment":"The word 'identificaition' appears twice in the proof of Lemma 4.2 and should be corrected to 'identification'.","section":"Section 4, proof of Lemma 4.2"},{"comment":"The expression (q;q)_infinity^{-1-(m+t_1+t_2)m/(t_1 t_2)} involves negative and non-integer exponents. Since the paper works with formal power series, a brief sentence stating that these powers are interpreted in Q(t_1,t_2)[[q]] would prevent ambiguity.","section":"Section 5, equation (5.16)"},{"comment":"The notation t_1^{delta(.)} is defined through Definition 3.4(i), but the introduction uses it before that definition is stated. An example of the action of t_1^{delta(.)} on a generalized partition would improve readability.","section":"Section 1, Theorem 1.2 and Conjecture 1.3"},{"comment":"The reference [Qin3] is listed only as 'Preprint' with no date or availability information. If it is intended as a companion paper, the authors should provide a public identifier or include the relevant statements in the present paper; otherwise the reader cannot assess the quoted results.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the unpublished companion preprint [Qin3], which is by one of the authors and carries the decisive qBD/qMZV membership and q-zeta expansion steps. I recommend that the editor require the authors to include the full statements and proofs of the quoted results, or to make the advertised Okounkov verification explicitly conditional, before the paper can be accepted. The vertex-operator identification in Theorem 1.2 appears solid and is the strongest part of the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the vertex-operator identification in Theorem 1.2—G_k equals t_2^k t_1^{δ(·)} B_k^{(α)} at α=−t_1/t_2 for k=0,1,2—is mostly self-contained and looks correct. The proof via the Jack/Macdonald eigenfunction lemma (Lemma 4.2) is clean, and the explicit G_2 formula (4.40) is a real new piece. Second, the partial verification of Okounkov's conjecture (Theorem 5.11) is not self-contained: the decisive qMZV membership and q-zeta expansions are imported from [Qin3], an unpublished preprint by one of the authors. If [Qin3]'s hypotheses don't match the coefficient extractions in (5.24)/(5.30), the conclusion doesn't follow from the text. That's the soft spot, and it is load-bearing.\n\nWhat the paper does well: the k=2 formula and the closed derivative formula in Theorem 6.4 are new and useful. The leading-term conjecture 1.3 is natural. The k=1 case recovers [OP1], and k=0 is trivial, so the novelty is narrow but solid. I did not machine-check every expansion, but the logic of Theorem 1.2 is transparent: Lemma 4.3 expands B(q,t^{-1}) to order 2, Lemma 4.4 derives the eigenvalue statement from the Jack limit, and Theorem 4.5 converts the eigenvalue statement to the operator identity. No fitted parameters, no circular reasoning in that part.\n\nThe concern the stress-test note raises is legitimate. Theorem 5.9 and 5.11 reduce the problem to explicit extractions and then invoke [Qin3] for membership in BD/qBD/qMZV, weight bounds, and the leading coefficient (−1)^k[k+2]. The present paper contains no proof of those claims. This is not a demonstrated error; it's a specific, addressable gap. It should be fixable—either by proving the needed statements in this paper or by posting [Qin3]—but until then the Okounkov verification is conditional.\n\nWho gets value: people working on Hilbert schemes and q-zeta values. The paper deserves a serious referee: the core vertex-operator result is worth publishing, and the conditional part can be handled by requesting the [Qin3] statements be made available or incorporated. I would not desk-reject; I would send it to review with a note to the referee to focus on the [Qin3] dependency and the G_2 computation.","headline":"Solid self-contained vertex-operator result for k≤2, but the Okounkov verification leans on an unpublished preprint and is conditional.","tokens_in":34189,"tokens_out":3035,"would_cite":true,"duration_ms":27973,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C05","11B65","17B69"],"pacs":[],"model":"deepseek-v4-flash","headline":"For degrees 0 through 2, the equivariant Chern character operator on Hilbert schemes of points in the affine plane equals a deformed-vertex operator, and this match partially verifies Okounkov's conjecture on q-analogues of multiple zeta…","keywords":["Hilbert schemes of points","equivariant Chern character operators","Okounkov's conjecture","multiple q-zeta values","Jack symmetric functions","Macdonald symmetric functions","deformed vertex operators","Heisenberg operators"],"falsifier":"Expand the explicit coefficient formula (5.30) for $N=1$ to compute the coefficient of $t_1^3$ in $\\langle\\mathrm{ch}_3\\rangle'_{0,t_1,t_2}$. The theorem predicts this coefficient is $- [5]$ plus a finite linear combination of lower-weight elements of $\\mathrm{qMZV}$; finding a term outside $\\mathrm{qMZV}$, or a leading bracket different from $-[5]$, would disprove the partial verification.","tokens_in":33155,"feed_emoji":"🧮","tokens_out":10693,"duration_ms":90338,"temperature":0.7,"pith_summary":"The paper aims to identify the equivariant Chern character operators $G_k(t_1,t_2)$ acting on the equivariant cohomology of Hilbert schemes of points in $\\mathbb{C}^2$ with concrete vertex-operator expressions, and to use that identification to make progress on Okounkov's conjecture that certain reduced series are $q$-analogues of multiple zeta values. The main theorem proves the identification for $k=0,1,2$: $G_k(t_1,t_2)=t_2^k t_1^{\\delta(\\cdot)}B_k^{(\\alpha)}|_{\\alpha=-t_1/t_2}$, where the $B_k^{(\\alpha)}$ are coefficients of an operator built from the deformed vertex operator introduced in [CW]. A direct consequence is that Okounkov's reduced single-operator series at $m=0$ lies in the algebra $\\mathrm{qMZV}[t_1,t_2]$, with leading coefficient $(-1)^k[k+2]$; the multi-operator series lies in the larger algebra $\\mathrm{qBD}[t_1,t_2]$. A sympathetic reader would care because the result connects cup-product geometry on Hilbert schemes to the combinatorics of symmetric functions and to concrete arithmetic $q$-series.","feed_headline":"Chern character operators equal vertex operators for k up to 2","feed_subtitle":"The match partially proves Okounkov's conjecture: reduced series are q-multiple zeta values.","key_machinery":"The load-bearing object is the operator $B(q,t^{-1})$ formed from the zero mode of the deformed vertex operator $V(z;q,t,\\tilde q,\\tilde t)=\\exp\\big(\\sum_{k\\ge1}(q^k-\\tilde q^k)z^k a_{-k}/k\\big)\\exp\\big(\\sum_{k\\ge1}(\\tilde t^k-t^k)z^{-k}a_k/k\\big)$, normalized by $((1-q)(1-t^{-1}))^{-1}$. The paper proves in Lemma 4.2 that the integral form $J_\\lambda(X;q,t)$ of the Macdonald symmetric function is an eigenfunction of this operator with eigenvalue $\\sum_{\\square} q^{a'(\\square)}t^{-\\ell'(\\square)}$. Because this eigenvalue is exactly the equivariant Chern character weight at the torus-fixed point $\\xi_\\lambda$ of the Hilbert scheme, the operator $t_2^k t_1^{\\delta(\\cdot)}B_k^{(\\alpha)}|_{\\alpha=-t_1/t_2}$ reproduces $G_k$ up to degree $k=2$. The same operator calculus then feeds the trace computations via the Ext vertex operators $\\Gamma_\\pm(z)$, turning reduced series into coefficient extractions of $q$-bracket expressions.","core_discovery":"On its own terms, the paper establishes Theorem 1.2: for $0\\le k\\le 2$, the operator $G_k(t_1,t_2)$ defined by equivariant cup product with the $k$-th equivariant Chern character of the tautological bundle equals $t_2^k t_1^{\\delta(\\cdot)} B_k^{(\\alpha)}$ evaluated at $\\alpha=-t_1/t_2$. Here $B(q,t^{-1})$ is the operator $\\frac{1}{(1-q)(1-t^{-1})}(1-V_0(z;t,t^{-1},1,qt^{-1}))$, with $V$ the deformed vertex operator introduced in [CW], and $B_k^{(\\alpha)}$ is the coefficient of $t_0^k$ after setting $q=e^{\\alpha t_0}$, $t=e^{t_0}$. The proof passes through the Fock-space identification of $H'_X$ with symmetric functions: the integral form $J_\\lambda(X;q,t)$ of the Macdonald polynomial is an eigenfunction of $B(q,t^{-1})$ with eigenvalue $\\sum_{\\square\\in D_\\lambda}q^{a'(\\square)}t^{-\\ell'(\\square)}$, exactly the localization eigenvalue of the equivariant Chern character at the fixed point $\\xi_\\lambda$. With this identification in hand, the paper derives the partial verification Theorem 1.4: $\\langle\\mathrm{ch}_k\\rangle'_{0,t_1,t_2}\\in \\mathrm{qMZV}[t_1,t_2]$ is a degree-$k$ symmetric homogeneous polynomial whose coefficients have weight at most $k+2$, and the coefficient of $t_1^k$ is $(-1)^k[k+2]$ plus lower-weight terms. For several operators with $k_i\\in\\{0,1,2\\}$ and arbitrary $m$, the reduced series lies in $\\mathrm{BD}(t_1,t_2)[m]$.","pith_inferences":["An extension the authors do not pursue is to test Conjecture 1.3 for $k=3$ by computing the leading term of $G_3$ through the same vertex-operator expansion; the paper notes that the exact identity (1.12) is unlikely to hold beyond $k=2$, so only the leading-term comparison is expected to survive.","If the companion results quoted as [Qin3] are written out and checked against (5.24) and (5.30), the same coefficient extraction may upgrade Theorem 1.4 from the single-operator $m=0$ case to the full Okounkov conjecture by tracking weights through the vertex-operator expansion.","The eigenfunction mechanism suggests a broader dictionary: any operator on $H'_X$ that is diagonal on the fixed-point basis with weights of the form $\\sum_{\\square} f(a'(\\square),\\ell'(\\square))$ may be realizable as a zero mode of a deformed vertex operator, giving a constructive route to the higher Chern character operators."],"forward_implications":["For a single Chern character operator, the reduced series at $m=0$ is a polynomial in $t_1,t_2$ with coefficients in $\\mathrm{qMZV}$, so Okounkov's conjecture holds in that case (Theorem 1.4(ii)).","For $k_i\\in\\{0,1,2\\}$, the full reduced series at arbitrary $m$ lands in $\\mathrm{BD}(t_1,t_2)[m]$, meaning it is a polynomial in $m$ whose coefficients are bi-brackets.","The explicit formula (4.40) for $G_2$ gives the first beyond-boundary equivariant Chern character operator in closed vertex-operator form, extending the known non-equivariant formula.","The leading-term structure of $G_k$ predicted in Conjecture 1.3 is consistent with the leading coefficients computed from the vertex-operator side, giving a concrete target for all $k$.","The commutation calculus with $G_1$ yields the leading term $n^k k!\\sum_{\\ell(\\lambda)=k+1,|\\lambda|=-n}(-1)^{\\ell(\\lambda_+)}(t_1t_2)^{\\ell(\\lambda_-)-1}a_\\lambda/\\lambda!$ for the higher derivatives $a_{-n}^{(k)}$ (Proposition 6.5)."],"supporting_citations":[{"why":"Introduces the deformed vertex operator $V(z;q,t,\\tilde q,\\tilde t)$ from which the operator $B$ is built.","marker":"[CW]"},{"why":"Defines the transformed Macdonald symmetric functions whose eigenvalue formula gives the action of $B(q,t)$.","marker":"[GH]"},{"why":"Supplies the corollary used in Lemma 4.2 for the eigenfunction statement for $J_\\lambda(X;q,t)$.","marker":"[Hai]"},{"why":"Provides the integral forms $J_\\lambda(X;q,t)$ and the Jack limit used to transfer eigenvalues to the equivariant Fock space.","marker":"[Mac]"},{"why":"States the conjecture on reduced series being $q$-analogues of multiple zeta values and defines the reduced series under study.","marker":"[Oko]"},{"why":"Constructs the Ext vertex operators $W(L,z)$ and the trace formula used to rewrite the reduced series.","marker":"[Car1]"},{"why":"Gives the vertex-operator identification for $W(L,z)$ via Exts, quoted as [Car1, Theorem 1].","marker":"[CO]"},{"why":"Supplies the decisive theorems that the coefficient expressions (5.24) and (5.30) lie in $\\mathrm{qBD}$/$\\mathrm{qMZV}$ with the stated weights.","marker":"[Qin3]"},{"why":"Defines the algebra inclusions $\\mathrm{qMZV}\\subset\\mathrm{qMD}\\subset\\mathrm{MD}$ used to interpret the series as $q$-multiple zeta values.","marker":"[BK1]"}],"fun_headline_variants":["Chern operators match vertex operators for k up to 2","Partial proof of Okounkov's conjecture via equivariant Chern characters","Equivariant Chern characters yield q-multiple zeta values","Okounkov conjecture partially proved for low degrees"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the unpublished results quoted as [Qin3] correctly establish the needed qBD/qMZV membership for the coefficient series appearing in (5.24) and (5.30); if those results are wrong or do not apply to these expressions, the paper's partial verification of Okounkov's conjecture does not follow from the arguments presented.","fun_headline_variants_meta":{"raw":{"variants":["Chern operators match vertex operators for k up to 2","Partial proof of Okounkov's conjecture via equivariant Chern characters","Equivariant Chern characters yield q-multiple zeta values","Okounkov conjecture partially proved for low degrees"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000928,"raw_usage":{"total_tokens":4030,"prompt_tokens":1058,"completion_tokens":2972,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":2903}},"tokens_in":674,"tokens_out":2972,"duration_ms":38090,"temperature":1.0,"reasoning_tokens":2903,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:31:56.047811+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Expand the explicit coefficient formula (5.30) for $N=1$ to compute the coefficient of $t_1^3$ in $\\langle\\mathrm{ch}_3\\rangle'_{0,t_1,t_2}$. The theorem predicts this coefficient is $- [5]$ plus a finite linear combination of lower-weight elements of $\\mathrm{qMZV}$; finding a term outside $\\mathrm{qMZV}$, or a leading bracket different from $-[5]$, would disprove the partial verification.","supporting_citations":[],"review_version":1}