论文标题
Hilbert-Poincaré系列的Matroid Chow环与交叉共同学
Hilbert-Poincaré series of matroid Chow rings and intersection cohomology
论文作者
论文摘要
我们研究了Matroids的Chow理论和Kazhdan-Lusztig框架中产生的四个物体的希尔伯特系列。这些分别是Chow环的Hilbert系列，增强的Chow环，相交的共同体学模块及其在空白公寓的茎。我们在曲霉的kazhdan-lusztig多项式与其盘子环的希尔伯特系列之间发展了一个并行性。这扩展到了$ z $ - 多种物质的矩形和希尔伯特系列的增强圈环之间的平行性。这建议将一个想法从一个框架带到另一个框架。我们的两个主要动机是所有这些多项式的实现猜想，以及计算它们的问题。我们提供了这些不变的几个内在定义。同样，通过利用它们在Matroid Polytope细分下的估值，我们推导了一种快速计算它们为大型矩阵计算它们的方法。统一的基金会是组合兴趣的情况。我们将所得的多项式与某些实用的家族（例如（二项式）Eulerian多项式）联系起来，我们解决了Hameister，Rao和Simpson的猜想。此外，我们通过Haglund和Zhang的结果证明了Hilbert系列的均匀成曲子的希尔伯特系列的真实性。此外，我们证明了Gedeon在Chow设置中的猜想的版本：均匀的矩阵最大化系数对具有固定等级和尺寸的矩形的多项式。通过依靠Kazhdan-Lusztig多项式的非负性以及Braden，Braden，Huh，Matherne，Proudfoot和Wang的半杂音分解，我们加强了希尔伯特（Hilbert多项式位于单型和实现之间；这解决了Ferroni，Nasr和Vecchi的猜想。
We study the Hilbert series of four objects arising in the Chow-theoretic and Kazhdan-Lusztig framework of matroids. These are, respectively, the Hilbert series of the Chow ring, the augmented Chow ring, the intersection cohomology module, and its stalk at the empty flat. We develop a parallelism between the Kazhdan-Lusztig polynomial of a matroid and the Hilbert series of its Chow ring. This extends to a parallelism between the $Z$-polynomial of a matroid and the Hilbert series of its augmented Chow ring. This suggests to bring ideas from one framework to the other. Our two main motivations are the real-rootedness conjecture for all of these polynomials, and the problem of computing them. We provide several intrinsic definitions of these invariants; also, by leveraging that they are valuations under matroid polytope subdivisions, we deduce a fast way for computing them for a large class of matroids. Uniform matroids are a case of combinatorial interest; we link the resulting polynomials with certain real-rooted families such as the (binomial) Eulerian polynomials, and we settle a conjecture of Hameister, Rao, and Simpson. Furthermore, we prove the real-rootedness of the Hilbert series of the augmented Chow rings of uniform matroids via a result of Haglund and Zhang; and in addition, we prove a version of a conjecture of Gedeon in the Chow setting: uniform matroids maximize coefficient-wisely these polynomials for matroids with fixed rank and size. By relying on the nonnegativity of the Kazhdan-Lusztig polynomials and the semi-small decompositions of Braden, Huh, Matherne, Proudfoot, and Wang, we strengthen the unimodality of the Hilbert series of Chow rings, augmented Chow rings, and intersection cohomologies to $γ$-positivity, a property for palindromic polynomials that lies between unimodality and real-rootedness; this settles a conjecture of Ferroni, Nasr, and Vecchi.
