{"id":"a6e9b278-2b69-4069-b6d1-7924eecb6977","arxiv_id":"2412.01283","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Data on Kazhdan-Lusztig polynomials up to S_11 suggests superexponential growth of extremal coefficients, near-universal unimodality, and a conjectured closed family (1+v+...+v^l)^{k-1}.","lead":"A team computed all Kazhdan-Lusztig polynomials for symmetric groups up to 11 strands and applied data-science tools to find patterns in them. The paper proposes several conjectures about how these polynomials grow and behave, including an explicit formula family that would imply superexponential growth.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Conjecture 5.4 is load-bearing for the superexponential growth claim, but its only justification is an unproved CP^1-to-CP^l extrapolation; a cheap check of the k=1, l=2 boundary case may already falsify the formula as stated.","rationale":"The reader's weakest assumption is exactly the CP^l extrapolation behind Conjecture 5.4, and I agree that this is the load-bearing premise for the strongest claim. I add a concrete boundary case that makes the test cheap and sharp: k=1 gives w^1_l=w0_{2l+1}, so l=2 predicts P_{1,w0_5}=1. The paper's data already include S_5 and the code is shipped, so this check is immediate and decisive for the formula as written. If the conjecture is intended only for k at least 2, that restriction must be stated, and the CP^l extrapolation would still lack proof. This does not change the reader's CONDITIONAL verdict: the paper is exploratory, transparent about the speculative status of Conjecture 5.4, and ships reproducible code and data, so the risk is already priced in. But the check should be run before Conjecture 5.4 is treated as reliable. I did not find a more central concern; Lemma 8.1's companion-matrix argument is doubtful, but it supports Conjecture 8.2 and is not the engine of the strongest claim.","tokens_in":18519,"tokens_out":16130,"duration_ms":146821,"concrete_test":"Run the provided Warrington KL program on w0_5=[5,4,3,2,1] and compare P_{1,w0_5} with 1; if P differs from 1, Conjecture 5.4 is false as stated for k=1,l=2 (n=5). Independently, compute P_{w^k_l} for the next cases (k,l)=(2,2),(3,2),(2,3) using the explicit one-line definitions in Section 5 and compare with (1+v+v^2)^{k-1} and (1+v+v^2+v^3)^{k-1}; this tests the CP^l extrapolation beyond the small-resolution regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Conjecture 5.4 is the engine behind Theorem 5.5(a) and hence behind the paper's superexponential growth claim (Conjecture 5.1). Its proof route is explicitly incomplete: [SSV98, Thm 2] works for l=1 because a small resolution exists, and the paper states after Thm 5.5 that for l>1 no such resolution exists and 'some other argument is needed.' That missing argument is not a formal gap in a theorem, but it is the entire support for the conjectured equality P_{w^k_l}=(1+...+v^l)^{k-1}. Moreover, the formula has a boundary case that the paper does not report checking. For k=1, the RHS is 1, and w^1_l is the longest permutation w0_{2l+1}; in particular w^1_2=w0_5 in S_5. The paper's own data (max evaluation 4 and max coefficient 2 for n=5) show nontrivial first-row KL polynomials exist at this rank, so whether P_{1,w0_5}=1 is a decisive, inexpensive test. If it fails, Conjecture 5.4 is false as stated; if the intended domain is k at least 2, that restriction is not stated and the CP^l extrapolation would still be unproved. Either way, the central superexponential-growth strategy rests on an unverified premise.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes the set of Kazhdan-Lusztig polynomials of symmetric groups S_n for n up to 11 using exploratory data analysis, statistics, and topological data analysis (ball mapper). It reports tables and visualizations for the density of nonzero polynomials, the number of distinct first-row polynomials, the growth of evaluations and maximal coefficients, unimodality, root locations, and Perron-Frobenius roots, and it formulates a series of conjectures and speculations. The main structural proposal is Conjecture 5.1 (superexponential growth of the maximal evaluation and maximal coefficient), for which Conjecture 5.4 gives a proposed family of permutations with Hilbert-Poincaré polynomials; Theorem 3.2 and Theorem 5.5(b) are the only fully external-theorem-backed results.","tokens_in":18858,"tokens_out":12647,"duration_ms":111217,"significance":"The paper ships a reproducible data repository, gives explicit falsifiable predictions, and bases its few theorems on external results rather than model fits. If Conjecture 5.4 held for an infinite family with k+2l=n, the Stirling bound in Theorem 5.5(a) would indeed imply superexponential growth, making the proposed family valuable. The data tables for S_11 are a useful resource. However, the central conjecture is currently supported mainly by an unproved CP^1-to-CP^l extrapolation, and the boundary case k=1, l=2 falsifies Conjecture 5.4 as stated; Lemma 8.1 also has an invalid proof. With these corrected, the exploratory contribution would be significant for the field.","major_comments":[{"comment":"Conjecture 5.4 is false as stated. For k=1, l=2 (n=5), the definition gives w^1_2 = w0_5, the longest element of S_5, while the right-hand side is (1+v+v^2)^0 = 1. The KL polynomial P_{1,w0_5} is not 1; direct computation gives P_{1,w0_5}=1+2v+v^2, consistent with the paper's own §5 table (max evaluation 4 and max coefficient 2 for n=5). Since Theorem 5.5(a) uses Conjecture 5.4 as the engine for Conjecture 5.1, this boundary failure removes the current support for the superexponential-growth strategy. If the intended family excludes k=1, that restriction is not stated and still leaves the CP^l extrapolation unproved.","section":"Section 5, Conjecture 5.4"},{"comment":"The proof of Lemma 8.1 is invalid. The displayed companion matrix has characteristic polynomial f, and its last column is (-b_0,...,-b_{d-1}); replacing those signs to obtain a nonnegative matrix changes the characteristic polynomial, so the Perron-Frobenius eigenvalue of the nonnegative matrix is not a negative root of f. The sentence 'taking the sign then back into account' does not bridge this gap. The probability estimate also treats the normalized coefficients b_i as independent uniform variables, which is not justified under the counting measure in the lemma. Note that the specific strong-connectivity objection in the proof is not the real issue: with b_0≠0 the companion graph is strongly connected; the sign step is the actual gap. Moreover, the §8 table shows PF percentages decreasing with n (set values 16.67, 21.43, 20, 15.36, 8.89, 3.81, 1.34 for n=5,...,11), which does not support Conjecture 8.2.","section":"Section 8, Lemma 8.1"},{"comment":"The text explicitly states that for l>1 'one does not have a small resolution of singularities, so some other argument is needed.' This missing argument is the entire justification for Conjecture 5.4 beyond the l=1 case, and it is load-bearing for Theorem 5.5(a). As it stands, the geometric analogy with [SSV98, Theorem 2] is not a proof strategy; the paper should present the l>1 case as an open problem with explicit checks (e.g., small l values) rather than as a conjecture whose truth would imply the headline growth statement.","section":"Section 5, after Theorem 5.5"}],"minor_comments":[{"comment":"The H polynomial is used without definition in the text; the reference to [BBD+22, Section 3.1] should be supplemented with a self-contained definition or at least the precise formula.","section":"Section 9"},{"comment":"The table following the root statistics has rows labeled '%' and 'av. real' without clear headers; the first row appears to be the percentage of roots with |root| in [0.9,1.1], but this should be stated explicitly.","section":"Section 7, table after root statistics"},{"comment":"The acknowledgment contains an unexplained knot PD code ('DT would like to thank the knot PD[X[3,1,4,32], ...]'); this should be removed or explained, as it is not relevant to the mathematical content.","section":"Acknowledgments"},{"comment":"The proof of Theorem 3.2 is only a citation to [HP08, Theorem 1.1]; for reproducibility, the derivation of the O(n^{-2}) bound from the cited theorem and the KL properties in Section 2 should be sketched.","section":"Theorem 3.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is exploratory and contains useful data and explicit conjectures, but the current version has two load-bearing defects: Conjecture 5.4 is false in an admissible boundary case, and Lemma 8.1's proof is invalid. Both are local and appear fixable by revision, so I do not recommend rejection; however, the claims that depend on these items must be reworded or supported. I would send the revised version back to the same referee."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a credibly honest big-data paper about KL polynomials. It ships code and data, most claims are explicitly labeled conjectures or speculations, and the theorem that is both nontrivial and properly load-bearing—the l=1 case from [SSV98]—is correctly attributed. The genuinely new item is Conjecture 5.4, the explicit family P_{w^k_l}=(1+v+...+v^l)^{k-1}, which if true gives superexponential growth of first-row coefficients and evaluations. I think that conjecture is worth a serious look, but the paper does not yet give it much support beyond an analogy with the l=1 small-resolution argument, and the text correctly admits that for l>1 no small resolution exists and \"some other argument is needed.\" That is not a flaw in a theorem; it is an honest caveat on a conjecture.\n\nWhat the paper does well: it is transparent about method, the dataset up to S_11 is real, the use of [HP08] and [SSV98] is appropriate, and the distinction between conjectures and speculations is useful. The unimodality and PF-root sections are clearly exploratory and presented as such.\n\nWhere I part company with the reader's report: the k=1 boundary check in the stress-test note is actually not a problem. w^1_l is the longest element w0_{2l+1}, and P_{e,w0}=1 because the full flag variety is smooth. So Conjecture 5.4 passes that cheap test, and the suggested falsifier does not land.\n\nThe real soft spots are two. First, the proof of Lemma 8.1 is wrong as written: for the companion matrix shown, strong connectivity fails already when the constant coefficient b0 vanishes, not merely when \"some\" b_i vanish, and the claim that the graph is strongly connected unless some b_i=0 is an if-and-only-if is false. The lemma may be salvageable, but the proof needs repair. Second, Conjecture 8.2 is in tension with the paper's own table: the percentage of PF polynomials among the set falls from 21.43 at n=6 to 1.34 at n=11, and the multiset percentage falls after n=9; a limit of 1 is not what those numbers suggest. The authors may believe finite-n effects dominate, but the trend is against them.\n\nBottom line: this is a paper for anyone working on KL polynomial asymptotics or on data-driven conjecturing in representation theory. It deserves a serious referee, but the referee should ask for the Lemma 8.1 proof to be fixed and for Conjecture 8.2 to be reconciled with the observed trend. I would accept it into peer review.","headline":"Honest, well-scoped data paper with one interesting new conjecture (5.4) and two genuine soft spots: a broken companion-matrix proof and a PF conjecture that runs against its own data table.","tokens_in":19427,"tokens_out":6842,"would_cite":true,"duration_ms":61275,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E10","62R07","20C08","68P05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper uses data from symmetric groups up to 11 strands to conjecture that a specific family of Kazhdan–Lusztig polynomials has the closed form $(1+v+\\cdots+v^l)^{k-1}$, which would make maximal coefficients grow superexponentially.","keywords":["Kazhdan–Lusztig polynomials","symmetric group","big data","topological data analysis","ball mapper","superexponential growth","Hilbert–Poincaré polynomial","unimodality"],"falsifier":"Check the first open case directly: compute $P_{w^7_2}$ for $S_{11}$, or $P_{w^3_2}$ for $S_7$, with the software used in the paper [War11], and compare all coefficients with $(1+v+v^2)^6$, respectively $(1+v+v^2)^2$. If any coefficient differs, Conjecture 5.4 is false; a match would support the formula but not prove the general statement.","tokens_in":18241,"feed_emoji":"📈","tokens_out":10556,"duration_ms":85782,"temperature":0.7,"pith_summary":"This paper tries to establish that Kazhdan–Lusztig polynomials of the symmetric group, a family of objects that is notoriously hard to compute and poorly understood, obey statistical regularities visible when all polynomials up to $S_{11}$ are collected and analyzed with data-science tools. The main target is an explicit formula: for the permutations $w^k_l$ with $k+2l=n$, the paper conjectures $P_{w^k_l}=(1+v+\\cdots+v^l)^{k-1}$, the Hilbert–Poincaré polynomial of $(\\mathbb{CP}^l)^{k-1}$. If true, the maximal coefficient and the evaluation at $v=1$ grow superexponentially in $n$, a sharp contrast with the exponential lower bound that follows from the proven $l=1$ case. The paper proves the formula for $l=0,1$ and leaves $l\\ge2$ as a conjecture, noting that the geometric argument used for $l=1$ does not extend directly. A sympathetic reader would care because explicit, provable formulas for Kazhdan–Lusztig polynomials are rare, and this one would pin down the true growth rate.","feed_headline":"Kazhdan–Lusztig coefficients conjectured to grow superexponentially","feed_subtitle":"Data from symmetric groups up to 11 strands points to a closed formula, proved so far only for l=0 and l=1.","key_machinery":"The machinery is the pair consisting of the permutation family $w^k_l$ and the Hilbert–Poincaré polynomial $(1+v+\\cdots+v^l)^{k-1}$ of $(\\mathbb{CP}^l)^{k-1}$, the cohomological count of a product of projective spaces. The paper interprets Kazhdan–Lusztig polynomials geometrically as Poincaré polynomials of intersection cohomology, so that the $l=1$ proof in [SSV98] works by resolving a singular flag variety and reducing the KL polynomial to ordinary cohomology, giving $(1+v)^b$ for a product of $b$ copies of $S^2\\simeq\\mathbb{CP}^1$. Conjecture 5.4 is exactly the assertion that replacing $\\mathbb{CP}^1$ by $\\mathbb{CP}^l$ preserves this geometric reduction, where the missing ingredient is the small resolution of singularities, a tool that smooths a singular space without changing its intersection cohomology. The supporting data toolkit—grid plots, effective exponents, root distributions, Perron–Frobenius-root tracking, and ball-mapper graphs—is used to detect and phrase the conjectures, not to prove them.","core_discovery":"The central claim is that the extremal Kazhdan–Lusztig polynomials of the symmetric group are governed by a clean product formula. For $k+2l=n$, let $w^k_l$ be the permutation that moves the first $l$ strands to the last $l$ positions, leaves $k$ middle strands fixed, moves the last $l$ strands to the first $l$ positions, and then stacks the longest elements of $S_l$ on both ends. The paper conjectures that $P_{w^k_l}=(1+v+\\cdots+v^l)^{k-1}$, the Hilbert–Poincaré polynomial of the product of $k-1$ copies of $\\mathbb{CP}^l$, and shows that this implies the maximal coefficient and the value at $v=1$ grow superexponentially in $n$ (Conjectures 5.1 and 5.4). The cases $l=0$ and $l=1$ are proved: the latter, via [SSV98, Theorem 2], gives $\\mathrm{ev}_n\\ge \\mathrm{coeff}_n\\in\\Omega(n^{-1/2}2^n)$. For $l\\ge2$, the conjecture stands open, because the small-resolution-of-singularities argument that proves the $l=1$ case has no analogue.","pith_inferences":["If Conjecture 5.4 holds, the family $w^k_l$ supplies a sharp benchmark for any method that estimates maximal Kazhdan–Lusztig coefficients, since the growth rate would be exactly that of the central multinomial coefficient.","Because the roots of $(1+v+\\cdots+v^l)^{k-1}$ all lie on the unit circle, the conjectured extremal polynomials would show that maximal coefficient growth can be superexponential while Perron–Frobenius roots stay small, consistent with Speculation 8.3.","The same data-analysis pipeline could be run on other Coxeter types or on $p$-canonical polynomials; a product formula analogous to Conjecture 5.4, if found, would indicate that the geometric mechanism is not special to type A."],"forward_implications":["If Conjecture 5.4 is correct, $\\mathrm{coeff}_n$ and $\\mathrm{ev}_n$ are in $\\Omega(\\gamma^n)$ for every $\\gamma>1$, because the central multinomial coefficient in $(1+v+\\cdots+v^l)^{k-1}$ grows superexponentially in $n$ according to Stirling's approximation.","The proven $l=0,1$ cases already give $\\mathrm{ev}_n\\ge \\mathrm{coeff}_n\\in\\Omega(n^{-1/2}2^n)$, so maximal Kazhdan–Lusztig coefficients grow at least exponentially in rank $n$.","The conjecture provides explicit extremal permutations whose Kazhdan–Lusztig polynomials are products of cyclotomic-type factors, giving a concrete test family for algorithms that compute these polynomials.","Combined with Conjecture 6.3, if almost all Kazhdan–Lusztig polynomials are unimodal, the superexponential growth of Conjecture 5.1 would be concentrated in a tiny minority of permutations."],"supporting_citations":[{"why":"Introduces the Kazhdan–Lusztig polynomials and the normalization and degree properties on which the paper's analysis relies.","marker":"[KL79]"},{"why":"Proves the $l=1$ case of Conjecture 5.4 by a small-resolution-of-singularities argument; this is the geometric template the conjecture seeks to generalize.","marker":"[SSV98]"},{"why":"Provides the program used to compute all Kazhdan–Lusztig polynomials in the paper's data set, including the $S_{11}$ data that drives the conjectures.","marker":"[War11]"},{"why":"Supplies the Bruhat-order density estimate used in Theorem 3.2, the paper's proven bound on the percentage of nonzero Kazhdan–Lusztig polynomials.","marker":"[HP08]"},{"why":"Provides the $H$-polynomials and combinatorial-invariance context used in the ball-mapper analysis and in relating the data to Bruhat intervals.","marker":"[BBD+22]"}],"fun_headline_variants":["Big data points to product formula for Kazhdan-Lusztig polynomials","Superexponential growth conjectured from KL data","Data-driven conjecture for Kazhdan-Lusztig polynomials","Product formula conjectured from Kazhdan-Lusztig data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conjecture depends on an unproved geometric transfer: the same kind of cohomological reduction that works for the $l=1$ case should work after replacing $\\mathbb{CP}^1$ by $\\mathbb{CP}^l$, even though the smoothing tool used for $l=1$ (a small resolution of singularities) does not exist for $l>1$.","fun_headline_variants_meta":{"raw":{"variants":["Big data points to product formula for Kazhdan-Lusztig polynomials","Superexponential growth conjectured from KL data","Data-driven conjecture for Kazhdan-Lusztig polynomials","Product formula conjectured from Kazhdan-Lusztig data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001177,"raw_usage":{"total_tokens":4815,"prompt_tokens":844,"completion_tokens":3971,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":460,"completion_tokens_details":{"reasoning_tokens":3902}},"tokens_in":460,"tokens_out":3971,"duration_ms":26946,"temperature":1.0,"reasoning_tokens":3902,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:32:04.741085+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the first open case directly: compute $P_{w^7_2}$ for $S_{11}$, or $P_{w^3_2}$ for $S_7$, with the software used in the paper [War11], and compare all coefficients with $(1+v+v^2)^6$, respectively $(1+v+v^2)^2$. If any coefficient differs, Conjecture 5.4 is false; a match would support the formula but not prove the general statement.","supporting_citations":[],"review_version":1}