Exercise 11, § 4

Let (W, S) be a Coxeter system of type H_3 (cf. Th. 1).

a) With the notation of Chap. V, § 6, no. 2. Proof of Lemma 2, show that $(z', z'') = \pi/5$; deduce that the Coxeter number h of W is equal to 10, and that the exponents of W are 1, 5 and 9.

b) Show, by using a), that Card(W) = 120, and that the number of reflections in W is 15.

c) Recover the formula Card(W) = 120 by applying Exerc. 5 of Chap. V, § 3.

d) Let $\mathscr{A}_5$ be the alternating group of {1, ..., 5}; if a, b, c, d are distinct elements of {1, ..., 5}, denote by (ab) the transposition of a and b, and (ab)(cd) the product of the transpositions (ab) and (cd). Let

Show that $(r_1 r_2)^5 = (r_2 r_3)^3 = (r_1 r_3)^2 = 1$. Deduce the existence of a homomorphism $f : W \to \mathscr{A}_5$ that takes S to $\{r_1, r_2, r_3\}$; show that f is surjective.

e) Let $\varepsilon : W \to \{\pm 1\}$ be the homomorphism $w \mapsto (-1)^{l(w)}$. Show that

is an isomorphism. (Use the fact that the two groups being considered have the same order.)