Grassmannians in the Lattice points of Dilations of the Standard Simplex

TL;DR

This paper explores the deep connection between the cohomology of Grassmannians and lattice points in dilated simplices, revealing new refined interpretations of classical polynomials through geometric and combinatorial lenses.

Contribution

It introduces novel refinements of the Ehrhart polynomial based on cohomological gradings, linking algebraic topology with combinatorial geometry in the context of Grassmannians.

Findings

Refined Ehrhart polynomials via cohomological gradings

Interpretation of Poincaré polynomial as lattice point count on hyperplanes

Establishment of a combinatorial-geometric correspondence

Abstract

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 $r\Delta_{d}$ 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 Poincar\'e polynomial ${\rm P(Gr}(d,d+r),z)$ 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 Poincar\'e polynomial of the Grassmannian Gr$(d,d+r)$ is a refinement of the Ehrhart polynomial $L_{\Delta_d}(r)$ of the standard $d$-simplex $\Delta_{d}$.