Exercise 2, § 4

Let R be a root system of type $E_6$.

a) Put $E = \mathbf{Q}(R)/3\mathbf{P}(R)$. This is an $\mathbf{F}_3$-vector space of dimension 5. With the notation of no. 12, show that the scalar product $(x|y)$ defines on E a non-degenerate symmetric bilinear form $\varphi$. Show that any two distinct elements of R have distinct images in E.

b) Let $\mathbf{O}(\varphi)$ be the orthogonal group of $\varphi$. Then, $\mathbf{O}(\varphi) = \{1, -1\} \times \mathbf{SO}(\varphi)$. The spinor norm defines a surjective homomorphism from $\mathbf{SO}(\varphi)$ to $\{1, -1\}$ whose kernel is denoted by $\mathbf{SO}^+(\varphi)$. The group $\mathbf{SO}^+(\varphi)$ is simple of order 25920. The quotient $\mathbf{O}(\varphi)/\mathbf{SO}^+(\varphi)$ is of type (2, 2). Deduce that $\mathbf{O}(\varphi)$ contains a unique subgroup $\Omega(\varphi)$ of index 2, distinct from $\mathbf{SO}(\varphi)$ and not containing $-1$.

c) Every element of $A(R)$ defines by passage to the quotient an element of $\mathbf{O}(\varphi)$. Show (by comparing orders) that this gives a homomorphism from $A(R)$ to $\mathbf{O}(\varphi)$. The image of $W(R)$ under this homomorphism is $\Omega(\varphi)$, and that of $W^+(R)$ is $SO^+(\varphi)$. Hence, $W(R)$ is an extension of $\mathbf{Z}/2\mathbf{Z}$ by a simple group of order 25920.

d) Let $F = Q(R)/2Q(R)$. This is an $\mathbf{F}_2$-vector space of dimension 6. Show that the quadratic form $\frac{1}{2}(x|x)$ defines by passage to the quotient a non-degenerate quadratic form $q_6$ on $F$, of pseudo-discriminant equal to 1. If $O(q_6)$ denotes the corresponding orthogonal group, define, by passing to the quotient, a homomorphism

Show that $h$ is injective (note that $-1 \notin W(R)$), and then that it is an isomorphism (compare orders). Deduce that there is an isomorphism from $W^+(R)$ to $O^+(q_6)$ (cf. Algebra, Chap. IX, § 9, no. 5).

e) By comparing c) and d), show$^1$ that $SO^+(\varphi)$ is isomorphic to $O^+(q_6)$.