Semi-direct Decompositions of Coxeter Groups

In this paper, we prove that any Coxeter group satisfying some conditions can be decomposed into a semi-direct product of two Coxeter groups. In (5), Weyl groups of some Kac-Moody algebras are decomposed into semi-direct products of Coxeter groups. In that paper, they introduce a concept of relative length, which is not easy to deal with at all. In our article, we generalize their result to any Coxeter group, using a rather fundamental method. That is, we prove that any Coxeter group, satisfying certain simplest conditions, can be decomposed into a semi-direct product of two Coxeter groups. Some decompositions are known and others are new. We hope our results will be helpful to the study of the structure theory and representation theory of some Coxeter groups, especially hyperbolic ones. This paper is organized as follows. In Section 1, we cite some notations and known results on Coxeter groups from (3). In Section 2, we prove our results in a special case, with some examples following. The results are generalized to a general case in Section 3. 1 Preliminaries