Semidirect product decomposition of Coxeter groups


Let (W,S)(W,S) be a Coxeter system, let S=I˙JS=I \dot{\cup} J be a partition of SS such that no element of II is conjugate to an element of JJ, let J~\widetilde{J} be the set of WIW_I-conjugates of elements of JJ and let W~\widetilde{W} be the subgroup of WW generated by J~\widetilde{J}. We show that W=W~WIW=\widetilde{W} \rtimes W_I and that (W~,J~)(\widetilde{W},\widetilde{J}) is a Coxeter system.Comment: 28 pages, one table. We have added some comments on parabolic subgroups, double cosets representatives, finite and affine Weyl groups, invariant theory, Solomon descent algebr

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