Asymptotical behaviour of roots of infinite Coxeter groups
Abstract
Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of "normalized" positive roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone of the bilinear form B associated to a geometric representation, and illustrate this property with numerous examples and pictures in rank 3 and 4. We also define a natural geometric action of W on E, and then we exhibit a countable subset of E, formed by limit points for the dihedral reflection subgroups of W. We explain that this subset is built from the intersection with Q of the lines passing through two positive roots, and finally we establish that it is dense in E.
Keywords
Cite
@article{arxiv.1112.5415,
title = {Asymptotical behaviour of roots of infinite Coxeter groups},
author = {Christophe Hohlweg and Jean-Philippe Labbé and Vivien Ripoll},
journal= {arXiv preprint arXiv:1112.5415},
year = {2019}
}
Comments
19 pages, 11 figures. Version 2: 29 pages, 11 figures. Reorganisation of the paper, addition of many details (section 5 in particular). Version 3 : revised edition accepted in Journal of the CMS. The number "I" was removed from the title since number "II" paper was named differently, see http://arxiv.org/abs/1303.6710