Reflections in abstract Coxeter groups

Abstract

Abstract. Let W be a Coxeter group and r ∈ 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 sugroups 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. IntroductionThe dihedral group of order 12 can be considered as Coxeter group of type I 2 (6) or as Coxeter group of type A 1 × I 2 (3). This example shows that, in general, the set of reflections in a Coxeter system is not determined by the abstract group W alone, but does indeed depend on the choice of the Coxeter generating set R. However there are a lot of examples of Coxeter systems (W, R) where the abstract group does determine the set of reflections or even the set R up to W -conjugacy. The main motivation for the present paper is to show that the latter holds for infinite Coxeter groups having a finite, irreducible and 2-spherical Coxeter generating set, which is our Theorem 1 below.In view of the main result of [5] it suffices to show that these Coxeter groups determine the set of reflections. In order to achieve this goal we provide a handy criterion for an involution in an abstract Coxeter group W to be a reflection with respect to any Coxeter generating set of W . Our principal observation is the following. Let (W, R) be a Coxeter system and let w ∈ W be an involution. If w ∈ R W , then the centralizer of w in W contains a finite normal subgroup properly containing w . This is an immediate consequence of Richardson's result in [16]. Thus, if w ∈ W is an involution having the property that w is a maximal finite normal subgroup of its centralizer in W , then w is a reflection with respect to any Coxeter generating set of W .It turns out that it is more convenient to work with the finite continuation of an involution rather than to consider finite normal subgroups of its centralizer. The finite continuation of a finite order element w in a Coxeter group is defined to be the intersection of all maximal finite subgroups containing it; we write FC(w) for the finite continuation of w. In this paper we restrict our attention to finitely generated Coxeter groups. For these it is a consequence of a result of Tits that every element of finite order is contained in some maximal finite subgroup; so FC(w) is a finite subgroup of W (see Corollary 14 below). The main result of the present paper is a complete description of the finite continuation of a simple reflection in a Coxeter system of finite rank. Its proof constitutes the bulk of this paper. Main Result: Let (W, R) be a Coxeter system of finite rank. Then the following holds.a) For each r ∈ R the finite continuation of r ...

On the Hurwitz action in finite Coxeter groups

“…Therefore parabolic subgroups in the classical sense coincide with parabolic subgroups in this case. As an immediate consequence of a theorem of Franzsen, Howlett and Mühlherr [FHM06] we also get the equivalence of the definitions for a large family of infinite Coxeter groups (including the irreducible affine Coxeter groups): Proposition 4.6. Let (W, S) be an infinite irreducible 2-spherical Coxeter system, that is S is finite, and ss ′ has finite order for every s, s ′ ∈ S. Then a subgroup of W is parabolic if and only if it is parabolic in the classical sense.…”

“…Clearly, if S is a simple system for (W, T ), then so is wSw −1 for any w ∈ W . Moreover, it is shown in [FHM06] that for an important class of infinite Coxeter groups including the irreducible affine Coxeter groups, all simple systems for (W, T ) are conjugate to one another in this sense. The following result is well-known and follows from [Dye90] Proposition 2.1.…”

“…Simple systems for (W, T ) have been studied by several authors (see [FHM06]). Clearly, if S is a simple system for (W, T ), then so is wSw −1 for any w ∈ W .…”

“…We adopt the approach and terminology introduced in [BDSW14] (see Section 2). In particular, we use an unusual definition of parabolic subgroup, which we show in Section 4 (after recalling some well-known facts on root systems in Section 3) to be equivalent to the classical definition for finite Coxeter groups and (as a consequence of results in [FHM06]) for a large family of irreducible infinite Coxeter groups, the so-called irreducible 2-spherical Coxeter groups. As a byproduct, we obtain some results on parabolic subgroups of finite Coxeter groups.…”

A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements

“…Such simple systems for (W, T ) were studied by several authors, see e.g. [FHM06] and the references therein. In particular, if S is a simple system for (W, T ) then so is wSw −1 for any w ∈ W .…”