Root Systems for Asymmetric Geometric Representations of Coxeter Groups |
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Authors: | Robert G. Donnelly |
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Affiliation: | 1. Department of Mathematics and Statistics , Murray State University , Murray, Kentucky, USA rob.donnelly@murraystate.edu |
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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. |
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Keywords: | Coxeter group Geometric representation Kac–Moody algebra Numbers game Root system Tits cone |
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