A positive formula for the Ehrhart-like polynomials from root system chip-firing
Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1159-1196.

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.

Received: 2018-04-09
Revised: 2019-04-01
Accepted: 2019-04-08
Published online: 2019-12-04
DOI: https://doi.org/10.5802/alco.79
Classification: 17B22,  52B20
Keywords: root system, chip-firing, Ehrhart polynomial, permutohedron, zonotope, root polytope
@article{ALCO_2019__2_6_1159_0,
     author = {Hopkins, Sam and Postnikov, Alexander},
     title = {A positive formula for the Ehrhart-like polynomials from root system chip-firing},
     journal = {Algebraic Combinatorics},
     pages = {1159--1196},
     publisher = {MathOA foundation},
     volume = {2},
     number = {6},
     year = {2019},
     doi = {10.5802/alco.79},
     mrnumber = {4049842},
     zbl = {07140429},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2019__2_6_1159_0/}
}
Hopkins, Sam; Postnikov, Alexander. A positive formula for the Ehrhart-like polynomials from root system chip-firing. Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1159-1196. doi : 10.5802/alco.79. https://alco.centre-mersenne.org/item/ALCO_2019__2_6_1159_0/

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