Combinatorics Seminar: Root system chip-firing

Seminar | November 20 | 12:10-1 p.m. | 939 Evans Hall

 Thomas McConville, MIT

 Department of Mathematics

The chip firing game is a simple example of a confluent system, which is a central notion in algebraic combinatorics and particle physics. That is, given an initial distribution of chips on a graph, there are many ways to "play" the chip firing game, but they all lead to the same result. Jim Propp introduced a variation on chip firing on the infinite line graph in which the chips are labeled. With this change, the game is no longer confluent from every initial configuration of chips, but it still exhibits some partial confluence. In an attempt to understand this partial confluence, we introduce a generalization called root system chip-firing. I will describe these processes and present several conjectures on the confluence of this game and on the positivity of some "Ehrhart-like" polynomials. Some parts of Alex's Colloquium talk on the same topic will be recalled, but the focus will be on new material. This is based on joint work with Sam Hopkins, Pavel Galashin, and Alexander Postnikov.