Exercise 6, § 4

a) Let $m$ be an integer $\geq 2$, or $+\infty$. Show that $4 \cos^2 \frac{\pi}{m} \in \mathbf{Z}$ is equivalent to $m \in \{2, 3, 4, 6, +\infty\}$.

b) Let $\Gamma$ be a lattice in $E$, i.e. a discrete subgroup of $E$ of rank $\dim E$. Show that, if $\Gamma$ is stable under $W$, the integers $m(s, s')$ for $s \neq s'$ all belong to the set $\{2, 3, 4, 6, +\infty\}$. (Observe that $\mathrm{Tr}(\sigma(w)) \in \mathbf{Z}$ for all $w \in W$; apply this result to $w = ss'$, and use a) above.)

c) Assume that $m(s, t) \in \{2, 3, 4, 6, , +\infty\}$ for $s \neq t \in S$. A family $(x_s)_{s \in S}$ of positive real numbers is called radical if it satisfies the following conditions:

If $(x_s)_{s \in S}$ is such a family, put $\alpha_s = x_s e_s$. Show that

Deduce that the lattice $\Gamma$ with basis $(\alpha_s)_{s \in S}$ is stable under $W$.

d) With the assumptions in c), suppose that the graph of $(W, S)$ is a forest. Show that in that case there exists at least one radical family $(x_s)$. (Argue by induction on Card(S); apply the induction hypothesis to S - {s_0}, where s_0 is a terminal vertex of the graph of S.)

e) With the assumptions in c), suppose that the graph of (W, S) is a circuit. Let n_4 (resp. n_6) be the number of arrows {s, t} of this graph whose coefficient m(s, t) is equal to 4 (resp. 6). Show that a radical family exists if and only if n_4 and n_6 are both even. If this condition is not satisfied, show that no lattice of E is stable under W (if S = {s_1, ..., s_n}, with s_i linked to s_{i+1} for 1 \leq i \leq n-1, and s_n linked to s_1, put c = s_1 ... s_n and show that Tr(\sigma(c)) is not an integer).