On the Todd Class of the Permutohedral variety

TL;DR

This paper derives a combinatorial formula for the Todd class of the permutohedral variety related to lattice point enumeration in polytopes, revealing its non-effectiveness in higher dimensions and positivity in Ehrhart polynomial coefficients.

Contribution

It provides a new combinatorial expression for the Todd class of the permutohedral variety and investigates its positivity properties.

Findings

The Todd class formula does not always have positive values for dimensions d ≥ 24.

The linear coefficient of the Ehrhart polynomial of any lattice generalized permutohedron is positive.

A combinatorial approach connects Todd classes with lattice point counting in polytopes.

Abstract

In the special case of braid fans, we give a combinatorial formula for the Berline-Vergne's construction for an Euler-Maclaurin type formula that computes number of lattice points in polytopes. Our formula is obtained by computing a symmetric expression for the Todd class of the permutohedral variety. By showing that this formula does not always have positive values, we prove that the Todd class of the permutohedral variety $X_d$ is not effective for $d\geq 24$. Additionally, we prove that the linear coefficient in the Ehrhart polynomial of any lattice generalized permutohedron is positive.