Root systems for asymmetric geometric representations of Coxeter groups

TL;DR

This paper explores the roots of asymmetric geometric representations of Coxeter groups, generalizing standard models to include Kac--Moody Weyl groups, and provides combinatorial characterizations of root properties and finiteness conditions.

Contribution

It introduces new characterizations of roots and finiteness conditions in asymmetric Coxeter group representations, extending known results to a broader class including Kac--Moody groups.

Findings

Characterization of when a multiple of a root is also a root

Conditions for finiteness of root multiples and positive roots sent to negative

Extension of symmetric case finiteness results to asymmetric representations

Abstract

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.