Unimodality of Kazhdan-Lusztig Polynomials of Matroids
Are the Kazhdan-Lusztig polynomials of matroids always unimodal - in particular log-concave, or even real-rooted, as conjectured? No: representable matroids obtained by deleting points from finite projective geometries have non-unimodal Kazhdan-Lusztig polynomials over every finite field, so the log-concavity and real-rootedness conjectures are both false.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Algebraic combinatorics
- Posed by
- Katie Gedeon, Nicholas Proudfoot, Benjamin Young
- Year posed
- 2017
- Years open
- 9y
- Solved
- 2026-07-27
- Model
- Rethlas agent (GPT-5.6 Sol)
- Vendor
- —
- Collaborators
- Ronnie Cheng, Shurui Liu
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The result came from running the Rethlas research agent with base model GPT-5.6 Sol at maximum reasoning effort - the same agent behind the Analytic Bertini entry; the paper documents the run and the authors verified the constructions.
Verification
arXiv preprint (v2) with explicit constructions from projective geometries; not yet peer-reviewed.
Source
arXiv:2607.24186 - Kazhdan-Lusztig polynomials of matroids need not be unimodal