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arXiv · 2607.24186

Kazhdan-Lusztig polynomials of matroids need not be unimodal

Ronnie Cheng·Shurui Liu
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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.

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Ronnie Cheng, Shurui Liu. 2026-07-28. Kazhdan-Lusztig polynomials of matroids need not be unimodal. https://arxiv.org/abs/2607.24186

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