Geometriae Dedicata

, Volume 110, Issue 1, pp 63–80

# Reflection Independence in Even Coxeter Groups

Article

## Abstract

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.

### Keywords

Coxeter group Coxeter system group presentation reflection simplex

## Preview

### References

1. 1.
2. 2.
Bahls, P.: A new class of rigid Coxeter groups, Internat. J. Algebra Comput. (in press).Google Scholar
3. 3.
Bahls, P.:Even rigidity in Coxeter groups, PhD Thesis, Vanderbilt University, 2002.Google Scholar
4. 4.
Charney, R., Davis, M. 2000When is a Coxeter system determined by its Coxeter group? JLondon Math. Soc.61441461Google Scholar
5. 5.
Kaul, A. 2002A class of rigid Coxeter groupsJ. London. Math. Soc.66592604Google Scholar
6. 6.
Mihalik, M. and Tschantz, S.: Visual decompositions of Coxeter groups, Preprint 2001.Google Scholar
7. 7.
Mihalik, M.: Classifying even Coxeter groups with non-even diagrams, Preprint, 2002.Google Scholar
8. 8.
Humphreys, J. 1990Reflection Groups and Coxeter Groups Cambridge, Stud. in Adv. Math. 29Cambridge University PressCambridgeGoogle Scholar
9. 9.
Bourbaki, N. 1981Groupes et algèbres de LieHermannParisChapter IV–VIGoogle Scholar
10. 10.
Carter, R. 1985Finite Groups of Lie Type: Conjugacy Classes and Complex CharactersWileyChichesterGoogle Scholar
11. 11.
Rotman, J. 1995An Introduction to the Theory of Groups, Grad. Texts in Math. 1484SpringerBerlinGoogle Scholar
12. 12.
Brady, N., McCammond, J., Mühlherr, B., Neumann, W. 2002Rigidity of Coxeter groups and artin groupsGeom. Dedicata.9491109Google Scholar