A positive formula for the Ehrhart-like polynomials from root system chip-firing

TL;DR

This paper proves a positive combinatorial formula for the coefficients of symmetric Ehrhart-like polynomials arising from root system chip-firing, advancing understanding of their structure and positivity.

Contribution

It provides the first positive combinatorial formula for symmetric Ehrhart-like polynomials from root systems, confirming part of a conjecture about their nonnegative coefficients.

Findings

Established a positive formula for symmetric Ehrhart-like polynomial coefficients

Linked the formula to properties of permutohedra and root polytopes

Suggested a conjecture for truncated Ehrhart-like polynomials, with limitations

Abstract

In earlier work in collaboration with Pavel Galashin and Thomas McConville we introduced a version of chip-firing for root systems. Our investigation of root system chip-firing led us to define certain polynomials analogous to Ehrhart polynomials of lattice polytopes, which we termed the symmetric and truncated Ehrhart-like polynomials. We conjectured that these polynomials have nonnegative integer coefficients. Here we affirm "half" of this positivity conjecture by providing a positive, combinatorial formula for the coefficients of the symmetric Ehrhart-like polynomials. This formula depends on a subtle integrality property of slices of permutohedra, and in turn a lemma concerning dilations of projections of root polytopes, which both may be of independent interest. We also discuss how our formula very naturally suggests a conjecture for the coefficients of the truncated Ehrhart-like polynomials that turns out to be false in general, but which may hold in some cases.