Exercise 12, § 4

Let (1, i, j, k) be the canonical basis of the field H of quaternions, by means of which H is identified with $\mathbf{R}^4$. Equip H with the scalar product $\frac{1}{2}(x \bar{y} + y \bar{x})$. Let $\Gamma$ be the multiplicative group of quaternions of norm 1.

a) If $a \in \Gamma$, the orthogonal reflection $s_a$ in H which transforms a to -a is the map $x \mapsto -a \bar{x} a$.

b) Let $q = \cos \frac{\pi}{5} - \frac{1}{2}i + (\cos \frac{3\pi}{5})j \in \Gamma$, and $r = \frac{1}{2}(1 + i + j + k) \in \Gamma$. Let Q be the set of quaternions obtained from 1, q, r by arbitrary even permutations and sign changes of the coordinates. Then Q is a subgroup of $\Gamma$ of order 120.

c) Let W be the subgroup of $\mathbf{GL}(H)$ generated by the $s_a$ for $a \in Q$. Show that W leaves Q stable, is finite, irreducible, and non-crystallographic; deduce that W is of type H_4 (cf. Theorem 1).

d) Show that W acts transitively on Q (use Prop. 3 of Chap. IV, § 1).

e) Let $a_0 \in Q$, let V be the vector subspace of H orthogonal to $a_0$, and let $W_0$ be the stabiliser of $a_0$ in W. Show that the restriction of $W_0$ to V is an irreducible group generated by reflections (Chap. V, § 3, no. 3) and that it is non-crystallographic. Deduce that $W_0$ is a Coxeter group of type H_3.

f) Show that Card(W) = $2^6 3^2 5^2$ (use d), e) and Exerc. 11).

g) Show that the Coxeter number of W is equal to 30, and that its exponents are 1, 11, 19, 29.