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.
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Mathematics Subject Classifications (2000). 20F05, 20F55, 20F65, 51F15.
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Bahls, P., Mihalik, M. Reflection Independence in Even Coxeter Groups. Geom Dedicata 110, 63–80 (2005). https://doi.org/10.1007/s10711-003-1134-z
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DOI: https://doi.org/10.1007/s10711-003-1134-z