Semidirect product decomposition of Coxeter groups

Abstract

Let $(W,S)$ be a Coxeter system, let $S=I\dot{\cup }J$ be a partition of $S$ such that no element of $I$ is conjugate to an element of $J$, let $\tilde{J}$ be the set of $W_I$-conjugates of elements of $J$ and let $\tilde{W}$ be the subgroup of $W$ generated by $\tilde{J}$. We show that $W=\tilde{W}⋊W_I$ and that $(\tilde{W},\tilde{J})$ is a Coxeter system.

28 pages, one table. We have added some comments on parabolic subgroups, double cosets representatives, finite and affine Weyl groups, invariant theory, Solomon descent algebra