论文标题
标准单纯性膨胀晶格中的司法
Grassmannians in the Lattice points of Dilations of the Standard Simplex
论文作者
论文摘要
A remarkable connection between the cohomology ring ${\rm H^{\ast}(Gr}(d, d+r),\Z)$ of the Grasssmannian ${\rm Gr}(d,d+r)$ and the lattice points of the dilation $rΔ_{d}$ of the standard d-simplex is investigated.同时学的自然分级诱导$rδ_{d} $的晶格点的不同等级。这导致了Ehrhart多项式$ L_ {δ_{D}}（r）$的不同改进。我们研究了由权重$（1,1，\ dots，1）$和$（1,2，\ dots，d）$定义的这两种精炼。其中一种改进将庞加莱多项式$ {\ rm p（gr}（d，d+r），z）$解释为晶格点的计数，该晶格点位于扩张的切片超平面上$rΔ_d$。因此，在组合级别上，格拉斯曼尼亚gr $（D，d+r）$的庞加莱多项式是Ehrhart多项式$ L_ {δ_d}（r）标准$ d $ -d $ -simplex $δ__{d} $的精致。
A remarkable connection between the cohomology ring ${\rm H^{\ast}(Gr}(d, d+r),\Z)$ of the Grasssmannian ${\rm Gr}(d,d+r)$ and the lattice points of the dilation $rΔ_{d}$ of the standard d-simplex is investigated. The natural grading on the cohomology induces different gradings of the lattice points of $rΔ_{d}$. This leads to different refinements of the Ehrhart polynomial $L_{Δ_{d}}(r)$ of the standard $d$-simplex. We study two of these refinements which are defined by the weights $(1,1,\dots,1)$ and $(1,2,\dots, d)$. One of the refinements interprets the Poincaré polynomial ${\rm P(Gr}(d,d+r),z)$ as the counting of the lattice points which lie on the slicing hyperplanes of the dilation $rΔ_d$. Therefore, on the combinatorial level the Poincaré polynomial of the Grassmannian Gr$(d,d+r)$ is a refinement of the Ehrhart polynomial $L_{Δ_d}(r)$ of the standard $d$-simplex $Δ_{d}$.
