Real-rootedness of matroid Kazhdan–Lusztig polynomials
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Are Kazhdan–Lusztig polynomials of matroids always real-rooted? The focused counterexample target is a representable rank-seven binary projective-deletion matroid. A later, separate direction concerns nonunimodal examples over every finite field.
Why it matters and what counts as progress
Why it matters: It distinguishes root constraints from coefficient unimodality, even within representable matroids.
What counts as progress: An explicitly representable matroid violating real-rootedness, or a theorem describing the classes where root or coefficient constraints survive.
How the problem developed
Katie Gedeon, Nicholas Proudfoot, and Benjamin Young Conjectured that every matroid Kazhdan–Lusztig polynomial has only negative real roots; real-rootedness was proved for certain families.
- 28 Jul 2026 · Counterexample [2]
Ronnie Cheng and Shurui Liu; Rethlas Constructed representable matroids over every finite field with non-unimodal Kazhdan–Lusztig polynomials, disproving the general real-rootedness conjecture as well as log-concavity.
AI contribution
Kazhdan-Lusztig polynomials of matroids need not be unimodal
Rethlas found a representable rank-seven binary projective-deletion matroid whose Kazhdan–Lusztig polynomial is not real-rooted, with a convolution and degree approach. Cheng and Liu later used that direction to construct representable matroids over every finite field with nonunimodal Kazhdan–Lusztig polynomials.
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AI-found matroid example seeds a stronger nonunimodality result