Breakthrough Tracker record
Claimed counterexamples to matroid Kazhdan–Lusztig real-rootedness and log-concavity
The authors construct representable matroids over every finite field whose Kazhdan–Lusztig polynomials are not even unimodal. The examples therefore refute the stronger conjectures that all such polynomials are log-concave or real-rooted.
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- Stable ID
math-matroid-kazhdan-lusztig-nonunimodal-2026- Revision
math-matroid-kazhdan-lusztig-nonunimodal-2026.v1- Field
- Mathematics · Combinatorics, algebraic geometry and matroid theory
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional claimed counterexamples
- Last checked
AI role
The authors say the Rethlas agent, using GPT-5.6 Sol at max reasoning effort, found the motivating real-rootedness counterexample. The paper records the prompt, accepted output, discovery trajectory, human proofs and a repair after the agent’s verifier rejected an earlier step.
Record details
- Problem or result
- Real-rootedness and log-concavity conjectures for Kazhdan–Lusztig polynomials of matroids
- Authors
- Ronnie Cheng and Shurui Liu
- Institutions
- Stanford University; the disclosed Rethlas research team
- Result date
- Preprint submitted July 27, 2026
Why it matters
The conjectures had survived extensive computation and proofs for many important matroid families. The projective-deletion construction gives counterexamples over every finite field and makes the failure stronger than non-real roots alone.
Limits
The manuscript is a first-version preprint and has not been conventionally peer reviewed. A specialist MathOverflow assessment reports that an independent counterexample was announced at ICM 2026 and verified, which supports the negative conclusion but does not review every argument in this paper.
Sources
- Primary: Cheng and Liu, Kazhdan–Lusztig polynomials of matroids need not be unimodal
- Independent specialist assessment: Luis Ferroni’s MathOverflow note on the paper and independently verified ICM counterexample
Correction and revision history
- 2026-07-29 — Added as a provisional claimed disproof after manuscript review and an independent specialist report of a separately verified counterexample with the same negative implication.
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