Repr\'esentations de r\'eflexion de groupes de Coxeter Cinqui\`eme partie: La repr\'esentation $R$ est r\'eductible. Cas particulier du rang $3$

Abstract

In this fith part, (with the notations of the preceding parts) we make the following hypothesis: $(W,S)$ is a Coxeter system, irreducible, $2$-spherical and $S$ is of cardinality $3$. Let $R:W\to GL(M)$ be a reducible reflection representation of $W$. Let $G:= Im\,R$. Each sub-space of $M$ $(\neq M)$ stabilize by $G$ is contained in $C_{M}(G)$. Let $M':=M/C_{M}(G)$ and $N(G):=\{g|g\in G,g\, \text{acts trivially on}\,M'\}$. We call $N(G)$ the translation sub-group of $G$. One of the goals of this part is to study $M'$ and $N(G)$. In particular it is shown that $N(G)$ is an $\mathcal{O}G-module$.