Grassmannians in the Lattice points of Dilations of the Standard Simplex

Abstract

A remarkable connection between the cohomology ring ${\mathrm{H}}^{\ast }(\mathrm{Gr}(d,d+r),\mathbb{Z})$ of the Grasssmannian $\mathrm{Gr}(d,d+r)$ and the lattice points of the dilation $r{\Delta }_d$ of the standard d-simplex is investigated. The natural grading on the cohomology induces different gradings of the lattice points of $r{\Delta }_d$. This leads to different refinements of the Ehrhart polynomial $L_{{\Delta }_d}(r)$ 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 $\mathrm{P}(\mathrm{Gr}(d,d+r),z)$ as the counting of the lattice points which lie on the slicing hyperplanes of the dilation $r{\Delta }_d$. Therefore, on the combinatorial level the Poincaré polynomial of the Grassmannian Gr$(d,d+r)$ is a refinement of the Ehrhart polynomial $L_{{\Delta }_d}(r)$ of the standard $d$-simplex ${\Delta }_d$.