Exercise 4, § 4

Let $R$ be an irreducible reduced root system in $V$. $(\alpha_1, \ldots, \alpha_l)$ a basis of $R$, $\tilde{\alpha}$ the highest root. Put $\tilde{\alpha} = n_1 \alpha_1 + \cdots + n_l \alpha_l$. We are going to determine the maximal closed, symmetric subsets of $R$, distinct from $R$.

a) Let $i \in \{1, 2, \ldots, l\}$. Let $R_i$ be the set of $\alpha \in R$ which are linear combinations of the $\alpha_j$ for $j \neq i$. Show that $R_i$ is maximal if and only if $n_i = 1$.

(1) M. Kneser has shown that all the "exceptional isomorphisms" between the finite classical groups can be obtained by using an analogous method. Cf. Über die Ausnahme-Isomorphismen zwischen klassischen Gruppen, Hamburger Abh., Vol. XXXI (1967), p. 136-140.

b) Let $i \in \{1, 2, \ldots, l\}$ and assume that $n_i > 1$. Let $S_i$ be the set of roots $\sum_{j=1}^l m_j \alpha_j$ with $m_i \equiv 0 \pmod{n_i}$. Show that $S_i$ is maximal if and only if $n_i$ is prime. (If $n_i = ab$ with $a > 1, b > 1$, consider the subset $S'$ of $R$ consisting of the roots $\sum_j m_j \alpha_j$ with $m_i \equiv 0 \pmod{a}$, and show that $S'$ strictly contains $S_i$.) Show that the roots $-\tilde{\alpha}, \alpha_j \ (j \neq i)$ form a basis of $S_i$ (which is of rank $l$). Deduce the Dynkin graph of $S_i$.

c) Every maximal closed, symmetric subset of $R$ is transformed by an element of $W(R)$ into one of the subsets described in $a$ or $b$. (Let $\Sigma$ be such a subset. Then $\Sigma = R \cap H$, where $H$ is a subgroup of $Q(R)$ of finite index ($§ 1$, Exerc. 6 b)); we can assume that $Q(R)/H$ is cyclic. Then there exists $u^* \in V^*$ such that $\Sigma$ is the set of $\alpha \in R$ such that $\langle u^*, \alpha \rangle \in \mathbf{Z}$.

d) List the maximal, closed, symmetric subsets of $R$ for the different types of irreducible reduced root systems.