Strong Convergence and a Game of Numbers

AbstractS. Mozes investigated a certain solitary game played on a weighted graph. Numbers are placed on the nodes of the graph, and a move consists of changing the sign of a negative number and changing the numbers on the neighboring nodes according to the weights on the edges. Mozes proved that the game has a strong convergence property when the edges have certain positive integer weights. However, his approach would give no information in the case of other weights. In this paper we first prove that strong convergence is equivalent to the fact that the game has as a certain ‘polygon property’. We can then, in a rather elementary way, characterize the assignments of weights that imply the polygon property, and hence strong convergence. Finally, we make a natural generalization of the game, where we also have weights on the nodes. The conditions for strong convergence generalize nicely to this game

Eriksson's numbers game and finite Coxeter groups

The numbers game is a one-player game played on a finite simple graph with certain ``amplitudes'' assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game and its interactions with Coxeter/Weyl group theory and Lie theory have been studied by many authors. In particular, Eriksson connects certain geometric representations of Coxeter groups with games on graphs with certain real number amplitudes. Games played on such graphs are ``E-games.'' Here we investigate various finiteness aspects of E-game play: We extend Eriksson's work relating moves of the game to reduced decompositions of elements of a Coxeter group naturally associated to the game graph. We use Stembridge's theory of fully commutative Coxeter group elements to classify what we call here the ``adjacency-free'' initial positions for finite E-games. We characterize when the positive roots for certain geometric representations of finite Coxeter groups can be obtained from E-game play. Finally, we provide a new Dynkin diagram classification result of E-game graphs meeting a certain finiteness requirement.Comment: 18 page

Root system chip-firing II: Central-firing

Jim Propp recently proposed a labeled version of chip-firing on a line and conjectured that this process is confluent from some initial configurations. This was proved by Hopkins-McConville-Propp. We reinterpret Propp's labeled chip-firing moves in terms of root systems: a "central-firing" move consists of replacing a weight $\lambda$ by $\lambda+\alpha$ for any positive root $\alpha$ that is orthogonal to $\lambda$. We show that central-firing is always confluent from any initial weight after modding out by the Weyl group, giving a generalization of unlabeled chip-firing on a line to other types. For simply-laced root systems we describe this unlabeled chip-firing as a number game on the Dynkin diagram. We also offer a conjectural classification of when central-firing is confluent from the origin or a fundamental weight.Comment: 30 pages, 6 figures, 1 table; v2, v3: minor revision