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arXiv · 0707.3310

Root systems for asymmetric geometric representations of Coxeter groups

Robert G. Donnelly
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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.

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Robert G. Donnelly. 2009-12-30. Root systems for asymmetric geometric representations of Coxeter groups. https://arxiv.org/abs/0707.3310

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