The numbers game, geometric representations of Coxeter groups, and Dynkin diagram classification results

Abstract

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 has been studied previously by Proctor, Mozes, Bjorner, Eriksson, and Wildberger. We show that those connected such graphs for which the numbers game meets a certain finiteness requirement are precisely the Dynkin diagrams associated with the finite-dimensional complex simple Lie algebras. As a consequence of our proof we obtain the classifications of the finite-dimensional Kac-Moody algebras and of the finite Weyl groups. We use Coxeter group theory to establish a more general result that applies to Eriksson's E-games: an E-game meets the finiteness requirement if and only if a naturally associated Coxeter group is finite. To prove this and some other finiteness results we further develop Eriksson's theory of a geometric representation of Coxeter groups and observe some curious differences of this representation from the standard geometric representation.