Root systems for asymmetric geometric representations of Coxeter groups

Details Root systems for asymmetric geometric representations of Coxeter groups Root systems for asymmetric geometric representations of Coxeter groups

2007-07-23

arxiv.org/abs/0707.3310v3

Mathematics

Group Theory

References updated; connections to the literature sharpened; some applications further developed. 15 pages, 1 figure

Scientific paper

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.

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