Abstract
Let (W, S) be a Coxeter system, let S = I ∪ J be a partition of S such that no element of I is conjugate to an element of J, let be the set of W I -conjugates of elements of J, and let be the subgroup of W generated by . We show that and that is the canonical set of Coxeter generators of the reflection subgroup of W. We also provide algebraic and geometric conditions for an external semidirect product of Coxeter groups to arise in this way, and explicitly describe all such decompositions of (irreducible) finite Coxeter groups and affine Weyl groups.
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