Root Systems for Asymmetric Geometric Representations of Coxeter Groups

Abstract

Results are obtained concerning root systems for 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 certain restrictions of all Kac–Moody Weyl groups. In particular, a characterization of when a nontrivial 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.

Key Words:

• Coxeter group
• Geometric representation
• Kac–Moody algebra
• Numbers game
• Root system
• Tits cone

2000 Mathematics Subject Classification:

• 20F55 (05E99)

ACKNOWLEDGMENTS

We thank Kimmo Eriksson for providing us with a copy of his thesis and for many helpful conversations during the preparation of this article. We thank Bob Proctor for his helpful feedback, which included the remarks concerning Weyl groups, and for sharing with us in advance some of the results of [22]. We thank Shrawan Kumar for helpful comments concerning Kac–Moody Tits cones.

Notes

Motivation for terminology: E-GCM's with integer entries are generalizations of ‘generalized’ Cartan matrices (GCM's), which are the starting point for the study of Kac–Moody algebras. Here we use the modifier “E” because of the relationship between these matrices and the combinatorics of Eriksson's E-games [11, 12].

In Proposition 6.9 of [11, 12] just prior to Proposition 4.4, it is asserted that s _x (Φ⁺∖{α _x }) = Φ⁺∖{α _x } for all x ∈ I _n . However, this will not be the case if Kα _x is a root for some K ≠ ±1. Only Theorem 6.9 of [11] and Proposition 4.4 of [12] are affected by this misstatement. (See Lemma 3.8 below.)

Communicated by K. Misra.