Eriksson's numbers game on certain edge-weighted three-node cyclic graphs

TL;DR

This paper investigates the divergence behavior of the numbers game on specific three-node cyclic graphs with real-valued amplitudes, contributing to the classification of E-game graphs related to Dynkin diagrams.

Contribution

It proves divergence of the numbers game on certain three-node cyclic graphs with real amplitudes, advancing the classification of E-game graphs with finiteness properties.

Findings

Numbers game diverges on certain three-node cyclic graphs with nonnegative initial values.

Divergence result is a key step in classifying E-game graphs related to Dynkin diagrams.

Supports the broader classification of graphs with finite E-game behavior.

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. Following Eriksson, we allow the amplitudes on graph edges to be certain real numbers. Games played on such graphs are ``E-games.'' We show that for certain such three-node cyclic graphs, any numbers game will diverge when played from an initial assignment of nonnegative real numbers not all zero. This result is a key step in a Dynkin diagram classification (obtained elsewhere) of all E-game graphs which meet a certain finiteness requirement.