Exercise 9, § 5

Let R be an irreducible root system, $R^{\vee}$ the inverse system, B a basis of R and $\alpha$ the highest root of R (relative to B). Put

$\alpha =\sum_{\beta\in B}n_{\beta}\beta$ and $\alpha^{\vee}=\sum_{\beta\in B}n^{\vee}_{\beta}\beta^{\vee}$;

let $\nu (R) =$ sup$_{\beta\in B}n^{\vee}_{\beta}$. a) Show that the interval $1, \nu (R)$ in $\mathbf{N}$ is the union of 1 and the $n^{\vee}_{\beta}$ for $\beta \in B$ (put $\alpha_0=\alpha$; let $(\alpha_1, . . . , \alpha_q)$ be a sequence of distinct elements of B such that $\alpha_i$ is not orthogonal to $\alpha_{i-1}$ for $i= 1, . . . , q$, such that $n^{\vee}_{\alpha}=\nu (R)$ and such that $q$ is maximal with these properties; prove that $n^{\vee}_{\alpha_i}=^{^q}i+ 1$ for $i= 1, . . . , q$).

b) Let $p$ be a prime number. Show that the following three properties are equivalent:

(i) $p\leq \nu (R)$;

(ii) there exists $\beta \in B$ such that $p=n^{\vee}_{\beta}$;

(iii) there exists $\beta \in B$ such that $p$ divides $n^{\vee}_{\beta}$.

In this case $p$ is called a torsion prime number of R.

c) For each type of irreducible root system, give the value of $\nu (R)$ and the torsion prime numbers. Show that the set of the $n_{\beta}$ and that of the $n^{\vee}_{\beta}$ coincide except for type $G_2$.

d) Let $R'$ be a symmetric closed subset of R that is irreducible as a root system. Show that $\nu (R')\leq \nu (R)$.