SEMIDIRECT PRODUCT DECOMPOSITION OF COXETER GROUPS

摘要

Let (W, S) be a Coxeter system, let S = I boolean OR J be a partition of S such that no element of I is conjugate to an element of J, let (J) over tilde be the set of W-1-conjugates of elements of J, and let (W) over tilde be the subgroup of W generated by (J) over tilde. We show that W = (W) over tilde (sic) W-1 and that (J) over tilde is the canonical set of Coxeter generators of the reflection subgroup (W) over tilde 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.