A positive formula for the Ehrhart-like polynomials from root system chip-firing
Name
1803.08472.pdf
Description
Accepted version
Size
903.63 KB
Format
Adobe PDF
Checksum (MD5)
f691e33bec44d475ea1a87b7b8d4f162
Author(s)
Hopkins, Sam
Postnikov, Alexander
Date Issued
2019
Journal
Algebraic Combinatorics
Publisher
Cellule MathDoc/CEDRAM
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.5802/ALCO.79
