Représentations de réflexion de groupes de Coxeter Cinquième partie: La représentation $R$ est réductible. 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 $G:=ImR$. Each sub-space of $M$ $(\ne M)$ stabilize by $G$ is contained in $C_M(G)$. Let $M^{\prime }:=M/C_M(G)$ and $N(G):=\{g|g\in G,g\text{acts trivially on}{M}^{\prime }\}$. We call $N(G)$ the translation sub-group of $G$. One of the goals of this part is to study $M^{\prime }$ and $N(G)$. In particular it is shown that $N(G)$ is an $\mathcal{O}G-module$.

23 pages, in French

References in corpus (1)

• Discrete complex reflection groups