## Christophe Hohlweg ; Jean-Philippe Labbé ; Vivien Ripoll - Asymptotical behaviour of roots of infinite Coxeter groups I

dmtcs:3088 - Discrete Mathematics & Theoretical Computer Science, January 1, 2012, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) - https://doi.org/10.46298/dmtcs.3088
Asymptotical behaviour of roots of infinite Coxeter groups I

Authors: Christophe Hohlweg ; Jean-Philippe Labbé ; Vivien Ripoll

Let $W$ be an infinite Coxeter group, and $\Phi$ be the root system constructed from its geometric representation. We study the set $E$ of limit points of "normalized'' roots (representing the directions of the roots). We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form associated to $W$, and illustrate this property with numerous examples and pictures in rank $3$ and $4$. We also define a natural geometric action of $W$ on $E$, for which $E$ is stable. Then we exhibit a countable subset $E_2$ of $E$, formed by limit points for the dihedral reflection subgroups of $W$; we explain how $E_2$ can be built from the intersection with $Q$ of the lines passing through two roots, and we establish that $E_2$ is dense in $E$.

Volume: DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)
Section: Proceedings
Published on: January 1, 2012
Imported on: January 31, 2017
Keywords: Coxeter group, root system, limit point, accumulation set.,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Fundings :
Source : OpenAIRE Research Graph
• Funder: Natural Sciences and Engineering Research Council of Canada