Reflections in abstract Coxeter groups

• Autores: W.N. Franzsen, R.B. Howlett, Bernhard Mühlherr
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 81, Nº 3, 2006, págs. 665-697
• Idioma: inglés
• DOI: 10.4171/cmh/69
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• Resumen

  • Let $W$ be a Coxeter group and $r\in W$ a reflection. If the group of order 2 generated by $r$ is the intersection of all the maximal finite subgroups of $W$ that contain it, then any isomorphism from $W$ to a Coxeter group $W'$ must take $r$ to a reflection in $W'$. The aim of this paper is to show how to determine, by inspection of the Coxeter graph, the intersection of the maximal finite subgroups containing $r$. In particular we show that the condition above is satisfied whenever $W$ is infinite and irreducible, and has the property that all rank two parabolic subgroups are finite. So in this case all isomorphisms map reflections to reflections.