Semidirect product decomposition of Coxeter groups
نویسنده
چکیده
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 J̃ be the set of WI -conjugates of elements of J and let W̃ be the subgroup of W generated by J̃ . We show that W = W̃ ⋊WI and that J̃ is the canonical set of Coxeter generators of the reflection subgroup W̃ 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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