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:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.79
Classification: 17B22,  52B20
Keywords: root system, chip-firing, Ehrhart polynomial, permutohedron, zonotope, root polytope
Hopkins, Sam 1; Postnikov, Alexander 1

1 Massachusetts Institute of Technology Department of Mathematics Cambridge MA 02139, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/articles/10.5802/alco.79/

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