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Big data approach to Kazhdan-Lusztig polynomials

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.01283 v3 pith:TFAI23OS submitted 2024-12-02 math.RT cs.LGmath.CO

Abel Lacabanne , Daniel Tubbenhauer , Pedro Vaz This is my paper · ORCID
classification math.RTcs.LGmath.CO MSC 05E1062R0720C0868P05
keywords Kazhdan–LusztigpolynomialssymmetricgroupbigdatatopologicalanalysisballmappersuperexponentialgrowthHilbert–Poincarépolynomialunimodality
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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 $l>1$.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Section 5, Conjecture 5.4] 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.
  2. [Section 8, Lemma 8.1] 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.
  3. [Section 5, after Theorem 5.5] 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.
minor comments (4)
  1. [Section 9] 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.
  2. [Section 7, table after root statistics] 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.
  3. [Acknowledgments] 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.
  4. [Theorem 3.2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central claims are conjectures or conditional implications backed by independent external results, not fitted inputs or load-bearing self-citations.

full rationale

The paper's derivation chain does not reduce to its own inputs. Theorem 3.2 is proved by quoting the external result [HP08, Theorem 1.1] and standard KL-polynomial facts, so the density bound is independent of the paper's data. Theorem 5.5(b) is proved for l=0 trivially and for l=1 by invoking the external [SSV98, Theorem 2]; the paper explicitly states that the l>1 case lacks a small resolution and that 'some other argument is needed,' so Conjecture 5.4 is presented as an open conjecture, not as a derived prediction. Theorem 5.5(a) is a conditional implication: 'Conjecture 5.4 implies Conjecture 5.1,' and the Stirling estimate is a standard external calculation. The self-cited repository [LTV24a] is used only as a location for code and data, and the other self-citation [LTV24b] appears in a future-directions paragraph rather than in any load-bearing argument. The many data-driven statements are explicitly labeled conjectures or speculations based on empirical patterns, and none is a fitted parameter renamed as a prediction. The skeptical concern about the unproved CP^1-to-CP^l extrapolation is a genuine correctness or completeness risk, but it is not circularity: the paper itself flags the missing proof, so no claimed derivation is being disguised as established mathematics.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No invented entities are introduced. The paper's derivations rely on standard KL polynomial properties, correctness of the cited computation, and two unproved bridges: the geometric argument behind Conjecture 5.4 and the companion-matrix argument behind Conjecture 8.2. The only hand-set free parameters are the ball-mapper radius and the unspecified exponent in Conjecture 3.1.

free parameters (2)
  • exponent a in den_n ~ n^{-(2+a)} = unspecified (conjectured 0 <= a <= 1)
    Conjecture 3.1 leaves the exponent a floating; the effective-exponent plot motivates the range, but no value is fitted.
  • ball mapper radius epsilon = 4e-9 for KL polynomials, 1e-9 for H polynomials
    Chosen by hand in Section 9; the authors claim the resulting shapes are stable across epsilon and n>=8, but the radius is a user-set parameter.
assumptions (4)
  • domain assumption Correctness of the KL polynomial computations produced by Warrington's program for S_n, n <= 11
    All data and conjectures depend on the computed KL polynomials; no independent verification is described (Remark 1A.1, Section 2(e)).
  • standard math Standard properties of KL polynomials listed in Section 2, including support, degree bound, and coefficient positivity
    Used throughout; these are cited from the standard literature [KL79, BB05, Lus03].
  • ad hoc to paper An analogue of [SSV98, Theorem 2] exists for l>1 with CP^l replacing CP^1, despite the absence of a small resolution of singularities
    This is the unproved geometric bridge behind Conjecture 5.4; the authors concede that 'some other argument is needed' after Theorem 5.5.
  • ad hoc to paper The companion-matrix Perron-Frobenius argument in Lemma 8.1 establishes that almost all f in 1+vZ[v] satisfy the negative PF property
    The proof as written appears invalid because the displayed matrix graph is not strongly connected for degree >1, so the standard Perron-Frobenius theorem does not apply as stated.

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Pith. "Pith review of Big data approach to Kazhdan-Lusztig polynomials." pith.science (2026). https://pith.science/paper/TFAI23OS

@misc{pith2026241201283,
  author       = {Pith},
  title        = {Pith review of: Big data approach to Kazhdan-Lusztig polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TFAI23OS}},
  note         = {Machine review of arXiv:2412.01283}
}
read the original abstract

We investigate the structure of Kazhdan-Lusztig polynomials of the symmetric group by leveraging computational approaches from big data, including exploratory and topological data analysis, applied to the polynomials for symmetric groups of up to 11 strands.

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Pith tools

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