AI 中文总结
研究在有限域上构造可表示拟阵，使其Kazhdan - Lusztig多项式非单峰，证明相关对数凹和实根猜想为假，通过从有限射影几何删点得例子，并给出收缩商半秩度条件下多项式性质。
AI 中文摘要
我们在每个有限域上构造了可表示拟阵，其Kazhdan - Lusztig多项式不是单峰的。特别是，关于拟阵的所有Kazhdan - Lusztig多项式是对数凹的以及它们有实根的猜想都是错误的。我们的例子是通过从有限射影几何中删除点得到的。更一般地，我们证明在每个收缩商的半秩度条件下，每个收缩的Kazhdan - Lusztig多项式枚举了其射影点位于相应删除集的子空间，而Z多项式与全射影几何的Z多项式一致。
英文摘要
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.
Comments17 pages
