Eriksson's numbers game and finite Coxeter groups

TL;DR

This paper explores the connections between Eriksson's numbers game, Coxeter groups, and Lie theory, focusing on finiteness properties, classification of initial positions, and geometric representations.

Contribution

It extends Eriksson's work by relating game moves to Coxeter group decompositions, classifies adjacency-free initial positions, and provides a new Dynkin diagram classification for finite E-games.

Findings

Extended Eriksson's work linking game moves to Coxeter group decompositions.

Classified adjacency-free initial positions using Stembridge's theory.

Provided a new Dynkin diagram classification for finite E-game graphs.

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 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.