Kazhdan-Lusztig polynomials of matroids need not be unimodal

Ronnie Cheng; Shurui Liu

Kazhdan–Lusztig polynomials of matroids need not be unimodal; counterexamples exist over every finite field.

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2026-07-31 21:31 UTC pith:DPWGWDOL

load-bearing objection Clean, high-confidence refutation of the log-concavity and real-rootedness conjectures for matroid KL polynomials, via an explicit projective-deletion theorem that holds up under both combinatorial and geometric proofs.

arxiv 2607.24186 v2 pith:DPWGWDOL submitted 2026-07-27 math.CO math.AG

Kazhdan-Lusztig polynomials of matroids need not be unimodal

Ronnie Cheng , Shurui Liu This is my paper

classification math.CO math.AG MSC 05B3505E1414F43

keywords Kazhdan–Lusztig polynomialsmatroidsunimodalitylog-concavityreal-rootednessprojective geometryZ-polynomialrepresentable matroids

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Two long-standing conjectures said that every matroid Kazhdan–Lusztig polynomial has log-concave coefficients and only real negative roots. This paper disproves both by building, over every finite field, representable matroids whose polynomials are not even unimodal. The examples come from deleting carefully chosen points from a finite projective geometry so that the Kazhdan–Lusztig polynomial simply counts the subspaces whose points all lie in the deleted set. Under a half-rank size condition on those deleted subspaces in every contraction, that count is exactly the Kazhdan–Lusztig polynomial, while the companion Z-polynomial stays the same as for the full geometry. Explicit choices then force a local valley in the coefficients, killing unimodality and therefore log-concavity and real-rootedness.

Core claim

Over every finite field F_q there exist F_q-representable matroids whose Kazhdan–Lusztig polynomials are not unimodal. More generally, if points are deleted from a projective geometry so that in every proper contraction the subspace-counting polynomial C_U has degree strictly less than half the quotient rank, then the Kazhdan–Lusztig polynomial of each contraction equals C_U and the Z-polynomial equals the Gaussian generating function of the full space. Choosing a six-dimensional deleted flat plus a suitable number of extra line-separated points produces an explicit non-unimodal polynomial, so the log-concavity and real-rootedness conjectures both fail.

What carries the argument

Projective deletions together with the half-rank degree condition: for the deleted set D, the subspace polynomial C_U enumerates subspaces whose projective points lie in the deleted set of the corresponding quotient; when deg C_U < (r − dim U)/2 for every proper flat span U, one obtains P_{M/F_U}(t) = C_U(t) and Z_M(t) equal to the full Gaussian binomial generating function.

Load-bearing premise

In every proper contraction, no collection of deleted points is allowed to contain a subspace that reaches half the dimension of that contraction; if any such large deleted subspace appears, the identification of the Kazhdan–Lusztig polynomial with the subspace count fails.

What would settle it

Build the explicit binary example of rank 17 obtained by deleting a 6-flat plus 589 line-separated points and compute its Kazhdan–Lusztig polynomial; the claimed coefficients are 1 + 652t + 651t^2 + 1395t^3 + 651t^4 + 63t^5 + t^6, which decrease then increase. Any mismatch, or any deleted configuration that secretly admits a degree-(r−dim U)/2 or larger subspace in some quotient, would refute the argument.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The log-concavity conjecture for matroid Kazhdan–Lusztig polynomials is false for representable matroids over every finite field.
The real-rootedness conjecture is likewise false; non-unimodal nonnegative coefficient sequences cannot be real-rooted.
For line-separated deletions of a k-flat plus s extra points (with k < r/2 and |D| small), the Kazhdan–Lusztig polynomial is exactly the truncated Gaussian polynomial of the k-flat plus an extra st term.
The Z-polynomial of every such deletion equals the Z-polynomial of the undeleted projective geometry.
Bose–Burton geometries appear as the undeleted special case and recover the known Gaussian Kazhdan–Lusztig polynomials when k < r/2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Any future positive conjecture about matroid Kazhdan–Lusztig polynomials must either restrict the matroid class (for example to paving, uniform, or graphic matroids) or replace unimodality by a weaker shape constraint that still allows local valleys.
The same deletion technique may produce counterexamples to other proposed coefficient inequalities once the half-rank condition is checked, because the polynomials become literally enumerative.
Because the examples are representable, the failure already occurs inside the geometric setting where the polynomials count intersection cohomology, not only for abstract matroids.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Clean, high-confidence refutation of the log-concavity and real-rootedness conjectures for matroid KL polynomials, via an explicit projective-deletion theorem that holds up under both combinatorial and geometric proofs.

read the letter

The headline is simple: over every finite field there are representable matroids whose Kazhdan–Lusztig polynomials are not unimodal, so both the EPW log-concavity conjecture and the GPY real-rootedness conjecture are false. That is the result people will remember.

What is actually new is Theorem 1.1. Under a half-rank degree bound on the subspace polynomials C_U in every proper contraction quotient, one gets P_{M/F_U}=C_U for every flat and Z_M equal to the Gaussian generating function of the ambient projective geometry. The line-separated deletions (Proposition 2.6) discharge the bound by a short counting-plus-complement argument, and the resulting family produces the explicit non-unimodal polynomials (k=6 plus a carefully chosen s). The Bose–Burton remark shows the construction is a controlled perturbation of a known family rather than an ad-hoc gadget.

The paper does the technical work cleanly. There are two independent proofs of the identification: a short induction from the characterizing axioms of P and Z, and a geometric argument that the forgetting map of matroid Schubert varieties is small precisely when the degree bound holds, so the decomposition theorem supplies the same equality. The arithmetic checks (Gaussian binomials, the binary rank-17 example, the cubic discriminant in the appendix) are correct. AI discovery of the initial rank-7 cubic is fully disclosed and does not carry the general theorems.

The only load-bearing hypothesis is the half-rank bound itself. It is verified for the constructions that matter; if someone later finds a deleted set that secretly admits a large subspace in some quotient, the identification would fail for that set, but that does not touch the counterexamples already written down. No circularity, no fitted parameters, citations are appropriate.

This is for anyone working on matroid KL polynomials, Z-polynomials, or the combinatorial side of intersection cohomology of matroid Schubert varieties. It deserves a serious referee and should be engaged with immediately.

Referee Report

0 major / 6 minor

Summary. The manuscript constructs, for every finite field F_q, an F_q-representable matroid whose Kazhdan–Lusztig polynomial is not unimodal, thereby disproving the log-concavity conjecture of Elias–Proudfoot–Wakefield and the real-rootedness conjecture of Gedeon–Proudfoot–Young. The examples are represented by P(V)\D, where D consists of a deleted projective subspace together with suitably separated points. Theorem 1.1 shows that, under the half-rank condition deg C_U(t)<(r-dim U)/2 for every proper flat span U, each contraction polynomial P_{M/F_U} equals an explicit subspace-enumerating polynomial C_U, while Z_M is the corresponding Gaussian-binomial polynomial. The paper gives both an inductive proof from the characterizing axioms and a geometric proof using matroid Schubert varieties, a birational small map, and the decomposition theorem. Corollary 1.3 then produces a strict coefficient valley over every finite field.

Significance. If accepted, this paper settles two prominent conjectures in matroid Kazhdan–Lusztig theory in the negative and substantially changes the expected coefficient behavior of these polynomials. Its strengths are notable: the counterexamples are explicit and representable over every finite field; the relevant polynomials are given by direct subspace counts; and the manuscript supplies two independent proofs of the main theorem, one combinatorial and one geometric via a small map and the decomposition theorem. The potentially delicate half-rank hypothesis is verified in Proposition 2.6 by a correct line-separation, complement, and cardinality argument. The explicit q=2 polynomial and the general q-binomial construction make the result readily checkable.

minor comments (6)

[§1] In the paragraph following axioms (i)–(iii), the sentence identifying Z_M is duplicated: “...is called the Z-polynomial of M. is called the Z-polynomial of M.” Please remove the repeated fragment.
[Corollary 1.2] “For every sufficiently larger” appears to be missing the variable r. It would also help to state the sufficient conditions explicitly—k<r/2, [k]_q+s<q^{r-2}, and s≤q^{r-k-1}—rather than only inside the proof.
[Proof of Proposition 2.6] The inference “Since every point of P(W) was deleted, one has U≰W” should explicitly use that U is a nonzero flat span: if U≤W, then E∩P(U) is empty and cannot span U.
[Remark 2.4] The specialization notation changes from t to q^m and later uses qm; please write q^m consistently and state at the first use that polynomial identity follows from equality at infinitely many m.
[§3] The letter k is reused for the base field after previously denoting dim W. A distinct symbol such as \Bbbk would avoid confusion. When invoking the small-map consequence of the decomposition theorem, it would also be useful to write the smallness criterion as codim{y:dim π^{-1}(y)≥i}>2i and note that finitely many strata and upper semicontinuity reduce it to the displayed inequalities.
[Appendix A.2] The introduction gives the rank-17 nonunimodal example, while the smaller rank-7 binary example with nonreal zeros appears only in the appendix. A brief forward pointer would make the relationship between the two examples clearer.

Circularity Check

0 steps flagged

No circularity: KL identification uses external uniqueness axioms; non-unimodality is an explicit Gaussian-binomial computation under a discharged degree bound.

full rationale

The paper’s central chain is not circular. Kazhdan–Lusztig polynomials are taken as the unique family satisfying the external axioms (i)–(iii) of [BV20] (degree bound and palindromicity of Z). Subspace polynomials C_U are defined by direct enumeration (1.1). Under the half-rank hypothesis deg C_U < (r−dim U)/2, induction plus the palindromic sum (2.2) forces P_{M/F_U}=C_U by uniqueness, not by redefinition. The hypothesis is verified for line-separated deletions by an independent counting/complement argument (Proposition 2.6), not assumed from the target. Non-unimodality is then a numerical comparison of Gaussian coefficients ([6 3]_q > [6 2]_q after the s-shift). The geometric Section 3 likewise invokes the external small-map consequence of BBD and known IH interpretations from EPW16/PXY18. Self-citations are ordinary background; nothing is fitted, and no load-bearing uniqueness theorem is imported from the authors’ own prior work. The derivation is self-contained against the stated axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 3 invented entities

The work rests on the standard axiomatic characterization of matroid Kazhdan–Lusztig and Z-polynomials, nonnegativity from singular Hodge theory, and classical finite-geometry counts. The only paper-specific ingredients are definitional (subspace polynomials, line-separated deletions, quotient deleted sets) and the half-rank degree hypothesis, which is discharged by direct counting in the constructions. No numerical parameters are fitted to data.

axioms (5)

domain assumption Kazhdan–Lusztig polynomials of matroids are uniquely characterized by P=1 in rank 0, deg P < rk/2, and palindromicity of the Z-polynomial Z_M(t)=∑_F t^{rk F} P_{M/F}(t) (EPW16; BV20 Thm 2.2).
Used as the inductive engine in the combinatorial proof of Theorem 1.1 and in Remark 2.4.
domain assumption Coefficients of P_M and Z_M are nonnegative for all matroids (BHMPW26 singular Hodge theory).
Cited for context; not required for the counterexamples themselves, which are nonnegative by direct count.
standard math Gaussian binomial coefficients count subspaces of F_q-vector spaces; standard facts on projective geometries and contractions of representable matroids.
Background counting used throughout §2.
domain assumption For representable matroids, P and Z record local and global intersection cohomology of matroid Schubert varieties; the decomposition theorem applies to small proper maps (EPW16, PXY18, BBD82).
Load-bearing only for the geometric proof in §3; the combinatorial proof is independent.
ad hoc to paper Half-rank degree condition deg C_U(t) < (r−dim U)/2 for every proper flat span U, verified for line-separated deletions with |D|<q^{r−2}.
The hypothesis that makes P_M=C_0; checked case-by-case in Proposition 2.6 rather than assumed abstractly for the counterexamples.

invented entities (3)

Subspace polynomial C_U(t) independent evidence
purpose: Enumerates subspaces A≥U whose projective points outside P(U) lie in the deleted set; identified with P_{M/F_U}.
Definition (1.1); the central dictionary of the paper.
Line-separated deletion sets independent evidence
purpose: Concrete deleted sets for which the half-rank bound is easy to check and C_0 is an explicit truncated Gaussian polynomial plus s t.
Definition 2.5 and Proposition 2.6; engineering device for counterexamples.
Quotient deleted sets D_U independent evidence
purpose: Show that contractions of projective deletions remain projective deletions, enabling induction.
Definition (2.3) and Lemma 2.3.

pith-pipeline@v1.2.0-grok45-kimik3 · 17764 in / 3201 out tokens · 68241 ms · 2026-07-31T21:31:19.597257+00:00 · methodology

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read the original abstract

We construct, over every finite field, representable matroids whose Kazhdan-Lusztig polynomials are not unimodal. In particular, the conjectures that all Kazhdan-Lusztig polynomials of matroids are log-concave and that they are real-rooted are both false. Our examples are obtained by deleting points from finite projective geometries. More generally, we prove that, under a half-rank degree condition in each contraction quotient, the Kazhdan-Lusztig polynomial of every contraction enumerates the subspaces whose projective points lie in the corresponding deleted set, while the $Z$-polynomial agrees with that of the full projective geometry.

Figures

Figures reproduced from arXiv: 2607.24186 by Ronnie Cheng, Shurui Liu.

**Figure 1.** Figure 1: Discovery process of Rethlas B. Paving-matroid relaxation. Formulas in [FNV23] produced formal hyperplane counts for which PM had nonreal zeros, but Rethlas showed that no matroid realized those counts. C. Deletion-interlacing. The deletion formula of Braden and Vysogorets [BV20] contains a negative term that prevents the standard interlacing argument and leads back to an open contraction-interlacing probl… view at source ↗

discussion (0)

Reference graph

Works this paper leans on

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