2026/07/27 by Ronnie Cheng, Shurui Liu · 1 voice
Mathematics · #math.AG #math.CO #msc:05A20 #msc:05B25 #msc:05B35 #msc:14F43
17 pages
arxiv created 2026/07/28 · arxiv updated 2026/07/30
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.