Semidirect product decomposition of Coxeter groups

TL;DR

This paper proves that a Coxeter group can be decomposed into a semidirect product of a subgroup generated by conjugates of a subset and another subgroup, with the latter forming a Coxeter system.

Contribution

It establishes a new decomposition of Coxeter groups into a semidirect product, identifying a Coxeter subsystem generated by conjugates of a subset.

Findings

Coxeter group decomposes as a semidirect product.

The subgroup generated by conjugates forms a Coxeter system.

Provides a structural insight into 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 $\widetilde{J}$ be the set of $W_I$-conjugates of elements of $J$ and let $\widetilde{W}$ be the subgroup of $W$ generated by $\widetilde{J}$. We show that $W=\widetilde{W} \rtimes W_I$ and that $(\widetilde{W},\widetilde{J})$ is a Coxeter system.