Root System Chip-Firing

Pavel Galashin ; Sam Hopkins ; Thomas McConville ; Alexander Postnikov

Séminaire lotharingien de combinatoire, 80B (2018)

Résumé

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.