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Asymptotical Behaviour of Roots of Infinite Coxeter Groups

Published online by Cambridge University Press:  20 November 2018

Christophe Hohlweg ,
Jean-Philippe Labbé  and
Vivien Ripoll
Christophe Hohlweg
Affiliation:
Université du Québec à Montréal,LaCIM et Département de Mathématiques, Montréal, Québec, H3C 3P8. e-mail: hohlweg.christophe@uqam.cavivien.ripoll@lacim.ca
Jean-Philippe Labbé
Affiliation:
Freie Universität Berlin, Institut für Mathematik, 14195, Berlin, Deutschland. e-mail: labbe@math.fu-berlin.de
Vivien Ripoll
Affiliation:
Université du Québec à Montréal,LaCIM et Département de Mathématiques, Montréal, Québec, H3C 3P8. e-mail: hohlweg.christophe@uqam.cavivien.ripoll@lacim.ca
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Abstract

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Let $W$ be an infinite Coxeter group. We initiate the study of the set $E$ of limit points of “normalized” roots (representing the directions of the roots) of $\text{W}$. We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form $B$ associated with a geometric representation, and we 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 how this subset is built fromthe intersection with $Q$ of the lines passing through two positive roots, and finally we establish that it is dense in $E$.

Keywords

17B2220F55Coxeter grouprootsroot systemlimit pointaccumulation set
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 2 , 01 April 2014 , pp. 323 - 353
Copyright
Copyright © Canadian Mathematical Society 2014

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