Geometriae Dedicata

, Volume 110, Issue 1, pp 63–80 | Cite as

Reflection Independence in Even Coxeter Groups



If (W,S) is a Coxeter system, then an element of W is a reflection if it is conjugate to some element of S. To each Coxeter system there is an associated Coxeter diagram. A Coxeter system is called reflection preserving if every automorphism of W preserves reflections in this Coxeter system. As a direct application of our main theorem, we classify all reflection preserving even Coxeter systems. More generally, if (W,S) is an even Coxeter system, we give a combinatorial condition on the diagram for (W,S) that determines whether or not two even systems for W have the same set of reflections. If (W,S) is even and (W,S′) is not even, then these systems do not have the same set of reflections. A Coxeter group is said to be reflection independent if any two Coxeter systems (W,S) and (W,S′) have the same set of reflections. We classify all reflection independent even Coxeter groups.


Coxeter group Coxeter system group presentation reflection simplex 


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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of Mathematics, 1326 Stevenson CenterVanderbilt UniversityNashvilleU.S.A.

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