On the Todd class of the permutohedral variety

Federico Castillo,Fu Liu

doi:10.5802/alco.157

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.

Highlights

Let Λ be a lattice of finite rank and V = Λ ⊗ R be the corresponding real finitedimensional vector space
A classical problem in the crossroads between enumerative combinatorics and discrete geometry is that of counting lattice points in lattice polytopes
What we want is a function that works simultaneously for all lattice polytopes. We can do this by requiring the function α to be local, i.e. the value of α(F, P ) only depends on the local geometry of P around F, or the value only depends on ncone(F, P ), Manuscript received 20th April 2020, revised and accepted 7th October 2020

Summary

Introduction

Despite of these positive results we’ve obtained in our previous work towards Conjecture 1.2, in the present paper we use our main result (the combinatorial formula described in Theorem 5.4) to find negative values for αbv on some cones in braid fans, disproving Conjecture 1.2 Note that this does not imply that Conjecture 1.1 is false, and we present a proof, independent of the rest of the paper, that the linear coefficient of the Ehrhart polynomial of any lattice generalized permutohedron is positive, providing further evidence to Conjecture 1.1. Let f be any square-free expression for Td(Xd) (as in Equation (8)), its symmetrization f is the Berline–Vergne expression for Td(Xd)

Combinatorial Tools

Computations in the Chow ring Ad

Formulas for the Berline–Vergne function

Positivity for linear coefficients