Example 4

Let V be a vector space over $\mathbf{Q}$ and R a root system in V; denote by P(R) the group of weights of R and by Q(R) the group of radical weights of R (Chap. VI, §1, no. 9). Then $(Q(R),0,R)$ and $(P(R),0,R)$ are root diagrams. A root diagram $(M,M_0,S)$ is isomorphic to a diagram of the form $(Q(R),0,R)$ (resp. $(P(R),0,R)$) if and only if M is generated by S (resp. $M^*$ is generated by $S^{\vee}$).

For every subgroup X of P(R) containing $Q(R), (X,0,R)$ is a root diagram and, up to isomorphism, every diagram $(M,M_0,S)$ such that $M_0= 0$, in other words such that S generates a subgroup of M of finite index, arises in this way.

The root diagram $(M,M_0,R)$ is said to be reduced if the root system R is reduced (in other words (Chap. VI, §1, no. 4) if the relations $\alpha , \beta \in R,\lambda \in \mathbf{Z}$, $\beta =\lambda \alpha$ imply that $\lambda = 1$ or $\lambda =-1$). The diagrams in Examples 1) and 3) are reduced.