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

Combinatorial Representation Theory

We attempt to survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. We give a personal viewpoint on the field, while remaining aware that there is much important and beautiful work that we have not been able to mention

From singularities to graphs

In this paper I analyze the problems which led to the introduction of graphs as tools for studying surface singularities. I explain how such graphs were initially only described using words, but that several questions made it necessary to draw them, leading to the elaboration of a special calculus with graphs. This is a non-technical paper intended to be readable both by mathematicians and philosophers or historians of mathematics.Comment: 23 pages, 27 figures. Expanded version of the talk given at the conference "Quand la forme devient substance : puissance des gestes, intuition diagrammatique et ph\'enom\'enologie de l'espace", which took place at Lyc\'ee Henri IV in Paris from 25 to 27 January 201