Root systems for asymmetric geometric representations of Coxeter groups

Results are obtained concerning the roots of asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include all Kac--Moody Weyl groups. In particular, a characterization of when a non-trivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting.Comment: References updated; connections to the literature sharpened; some applications further developed. 15 pages, 1 figur

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

Diamond-colored modular and distributive lattices

A modular or distributive lattice is `diamond-colored' if its order diagram edges are colored in such a way that, within any diamond of edges, parallel edges have the same color. Such lattices arise naturally in combinatorial representation theory, particularly in the study of poset models for semisimple Lie algebra representations and their companion Weyl group symmetric functions. Our goal is to gather in one place some elementary but foundational results concerning these lattice structures. Our presentation includes some new results as well as some new interpretations of classical results.Comment: 20 pages, 5 figure