Coxeter groups in Coxeter groups

Summary

Abstract

An automorphism of a Coxeter diagram M leads in a natural way to a Coxeter subgroup of the Coxeter group of type M. We introduce admissible partitions of Coxeter diagrams in order to generalize this situation. An admissible partition of a Coxeter diagram provides a Coxeter subgroup in a similar way. Our main result is a local criterion for the admissibility of a partition.

Introduction

We may ask in general which Coxeter groups arise as subgroups of a given Coxeter group. This question is of course far too general. However, there are Coxeter groups which arise canonically as subgroups of a given Coxeter group. Let for instance (W, S) be a Coxeter system and let S₁ be a subset of S, then (∧S₁), S₁) is again a Coxeter system.

Our purpose here is to introduce another way to obtain Coxeter subgroups in a given Coxeter group. In the example above we considered residues; the procedure, which will be treated here, has also a geometric background. We will deal with subcomplexes of the Coxeter complex which behave like subcomplexes fixed by a polarity. We do not go into the details concerning these geometric aspects. However, our procedure is motivated by the following consideration:

Let I be a set, let M be a Coxeter diagram over I and let (W, S) be the associated Coxeter system. Let l : W → N_o denote the length function.