Séminaire Lotharingien de Combinatoire, 80B.21 (2018), 12 pp.
Pavel
Galashin, Sam Hopkins, Thomas
McConville and Alexander
Postnikov
Root System Chip-Firing
Abstract.
Propp recently introduced a variant of chip-firing on the infinite
path where the chips are given distinct integer labels and conjectured
that this process is confluent from certain (but not all) initial
configurations of chips. Hopkins, McConville, and Propp proved Propp's
confluence conjecture. We recast this result in terms of root systems:
the labeled chip-firing game can be seen as a process which allows
replacing an integer vector λ by λ+α whenever
λ is orthogonal to α, for α a positive root of a
root system of Type A. We give conjectures about confluence for this
process in the general setting of an arbitrary root system. We show
that the process is always confluent from any initial point after
modding out by the action of the Weyl group (an analog of unlabeled
chip-firing in arbitrary type). We also study some remarkable
deformations of this process which are confluent from any initial
point. For these deformations, the set of weights with given
stabilization has an interesting geometric structure related to
permutohedra. This geometric structure leads us to define certain
"Ehrhart-like" polynomials that conjecturally have nonnegative
integer coefficients.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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