§ 4. Classification of root systems

1. Finite Coxeter groups

In this section, we are going to determine, up to isomorphism, all root systems, and consequently all crystallographic groups (§ 2, no. 5). More generally, we shall start by determining all finite groups generated by reflections in a finite-dimensional real vector space: this is equivalent (Chap. V, § 4, no. 8) to determining all finite Coxeter groups, or (Chap. V, § 4, no. 8, Th. 2) to determining all Coxeter matrices of finite order such that the associated bilinear form is positive and non-degenerate.

Let $M = (m_{ij})_{i,j \in I}$ be a Coxeter matrix of finite order $l$ . Put

q_{ij} = -\cos(\pi / m_{ij}).

Recall that $q_{ii} = 1$ and that $q_{ij} = q_{ji}$ is zero or $\leq -1/2$ for $i \neq j$ . Put $E = \mathbf{R}^l$ and let $(e_i)_{i \in I}$ be the canonical basis of $E$ . Denote by $(x|y)$ the bilinear form on $E$ associated to $M$ (Chap. V, § 4, no. 1) and $q$ the quadratic form $x \mapsto (x|x)$ on $E$ . For $x = \sum_{i \in I} \xi_i x_i$ ,

\| x \| ^2 = q(x) = \sum_{i,j \in I} q_{ij} \xi_i \xi_j.

Denote by $(X, f)$ the Coxeter graph of $M$ (Chap. IV, § 1, no. 9). If $a$ is an edge of $X$ , $f(a)$ is called the order of $a$ .

In the remainder of this number, the Coxeter group $W(M)$ defined by $M$ (Chap. V, § 4, no. 3) is assumed to be finite, so that $q$ is positive and non-degenerate and $X$ is a forest (Chap. V, § 4, no. 8, Prop. 8). We also assume that $X$ is connected (in other words that the Coxeter group $W(M)$ is irreducible), so that $X$ is a tree.

From the condition that $q$ is positive and non-degenerate, we shall obtain conditions on the $m_{ij}$ that will enable us to list all the possibilities for the corresponding Coxeter graphs; it will only remain to show that these possibilities are actually realised, in other words that the corresponding groups $W(M)$ are finite.

Lemma 1

For all $i$ , $\sum_{j \neq i} q_{ij}^2 < 1$ .

Let $J$ be the set of $j \in I$ such that $q_{ij} \neq 0$ , in other words such that $\{i, j\}$ is an edge of $X$ . If $j, j' \in J$ and $j \neq j'$ , $\{j, j'\}$ is not an edge (otherwise $i, j, j'$ would form a circuit), so $(e_j|e_{j'}) = 0$ . Let $F = \sum_{j \in J} \mathbf{R} e_j$ . Then $(e_j)_{j \in J}$ is an orthonormal basis of $F$ . The distance $d$ from $e_i$ to $F$ is given by $d^2 = 1 - \sum_{j \in J} (e_i|e_j)^2 = 1 - \sum_{j \in J} q_{ij}^2 = 1 - \sum_{j \neq i} q_{ij}^2$ , hence the lemma.

Lemma 2

Any vertex of $X$ belongs to at most 3 edges.

Indeed, if $i$ is linked to $h$ other vertices, the relations $q_{ij}^2 \geq \frac{1}{4}$ for these other vertices implies that $\frac{h}{4} < 1$ by Lemma 1, so $h \leq 3$ .

Lemma 3

If $i$ belongs to 3 edges, these edges are of order 3.

If not, we would have, in view of the relation $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ ,

\sum_{j \neq i} q_{ij}^2 \geq \frac{1}{4} + \frac{1}{4} + (\frac{\sqrt{2}}{2})^2 = 1

which is impossible (Lemma 1).

Lemma 4

If there exists an edge of order $\geq 6$ , then $l = 2$ .

Indeed, let $\{i, j\}$ be such an edge. If $l > 2$ , one of the edges $i, j$ (say $i$ ) would be linked to a third vertex $j'$ , since $X$ is connected. In view of the relation $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$ we would have

\sum_{k \neq i} q_{ik}^2 \geq \frac{1}{4} + (\frac{\sqrt{3}}{2})^2 = 1

which is impossible (Lemma 1).

Lemma 5

A vertex cannot belong to two distinct edges of order $\geq 4$ .

Let $i$ be such a vertex. We would have $\sum_{j \neq i} q_{ij}^2 \geq (\frac{\sqrt{2}}{2})^2 + (\frac{\sqrt{2}}{2})^2 = 1$ , which is impossible (Lemma 1).

Let $\{i, j\}$ be an edge of $X$ . We are going to define a new Coxeter graph, which will be obtained from the graph of $M$ by identifying $i$ and $j$ . The set $I'$ of its vertices is the quotient of $I$ obtained by identifying $i$ and $j$ . Put $p = \{i, j\}$ , which is an element of $I'$ , and identify the elements of $I$ distinct from $i$ and $j$ with their canonical images in $I'$ . Let $k, k'$ be two distinct elements of $I'$ . Then, $\{k, k'\}$ is an edge of the new graph in the following cases:

1) $k$ and $k'$ are distinct from $p$ and $\{k, k'\}$ is an edge of $X$ : in this case, the order of this edge is defined to be $m_{kk'}$ :

2) $k = p$ , and one of the sets $\{i, k'\}, \{j, k'\}$ is an edge of $X$ ; the order of $\{p, k'\}$ is defined to be $m_{ik'}$ if $\{i, k'\}$ is an edge of $X$ , and $m_{jk'}$ if $\{j, k'\}$ is an edge of $X$ (these two possibilities are mutually exclusive because $X$ is a tree).

Let $M' = (m'_{ij})_{i,j \in I'}$ be the new Coxeter matrix thus defined, and put $q'_{ij} = -\cos \frac{\pi}{m'_{ij}}$ . Then, for $k \neq p$ , $t'_{pk} = q_{ik} + q_{jk}$ . Thus, if $(\xi_i) \in \mathbf{R}^{I'}$ ,

\sum_{k, k' \in I'} q'_{kk'} \xi_k \xi_{k'} = \sum_{k, k' \in I} q_{kk'} \xi_k \xi_{k'} + \xi_p^2 - \xi_i^2 - \xi_j^2 - 2q_{ij} \xi_i \xi_j \tag{1}

= \sum_{k, k' \in I} q_{kk'} \xi_k \xi_{k'} - (1 + 2q_{ij}) \xi_p^2.

Lemma 6

If $\{i, j\}$ is of order 3, $W(M')$ is a finite Coxeter group.

Indeed, $q_{ij} = -\frac{1}{2}$ , so (1) becomes

\sum_{k,k' \in I'} q'_{kk'} \xi_k \xi_{k'} = \sum_{k,k' \in I} q_{kk'} \xi_k \xi_{k'}

and $(\xi_k)_{k \in I'} \mapsto \sum_{k,k' \in I'} q'_{kk'} \xi_k \xi_{k'}$ is a positive non-degenerate quadratic form. It now suffices to apply Th. 2 of Chap. V. § 4, no. 8.

Lemma 7

We have one of the following:

a) X has a unique ramification point (Chap. IV, Appendix, no. 1). and all the edges of X are of order 3. b) X is a chain and has at most one edge of order $\geq 4$ .

We argue by induction on l.

a) Assume that X has a ramification point i. Then i belongs to 3 edges of order 3. $\{i, k_1\}, \{i, k_2\}, \{i, k_3\}$ (Lemmas 2 and 3). If $l = 4$ the lemma is proved. If not, then $k_1$ , say, belongs to an edge distinct from those just mentioned since X is connected. Identify i and $k_1$ in the Coxeter graph of M. This gives a new graph to which the induction hypothesis can be applied, in view of Lemma 6. Now the image $p$ of i is a ramification point of the new graph $X'$ . Thus $X'$ has no other ramification point and all its edges are of order 3. Thus all the edges of X are of order 3, and i and $k_1$ are its only possible ramification points. But if $k_1$ were a ramification point of X, $p$ would belong to at least 4 edges in $X'$ , contrary to Lemma 2.

b) Assume that X has no ramification point. Then X is a chain (Chap. IV, Appendix, no. 3, Prop. 3). Let $\{i, j\}$ be an edge of order $\geq 4$ . If $l = 2$ , the lemma is trivial. If not, then i, say, belongs to an edge $\{i, k\}$ with $k \neq j$ (since X is connected). This edge is of order 3 (Lemma 5). Identify i and k in the Coxeter graph of M. By Lemma 6, the induction hypothesis can be applied. Let $p$ be the image of i in the new graph $X'$ . In $X'$ , $\{p, j\}$ is an edge of order $\geq 4$ , so $X'$ has no other edge of order $\geq 4$ , and hence $\{i, j\}$ is the only edge of order $\geq 4$ in X.

Lemma 8

Let $i_1, i_2, \ldots, i_p$ be vertices of X such that $\{i_1, i_2\}, \{i_2, i_3\}, \ldots, \{i_{p-1}, i_p\}$ are edges of order 3. Then $q(\sum_{r=1}^p r e_{i_r}) = \frac{1}{2} p(p+1)$ .

We have $(e_{i_r}|e_{i_r}) = 1, (e_{i_r}|e_{i_{r+1}}) = -\frac{1}{2}, (e_{i_r}|e_{i_s}) = 0$ if $s > r + 1$ . Thus

q(\sum_{r=1}^p r e_{i_r}) = \sum_{r=1}^p r^2 - 2 \sum_{r=1}^{p-1} \frac{1}{2} r(r+1) = p^2 - \sum_{r=1}^{p-1} r.

By Theory of Sets, Chap. III. § 5, no. 8. Cor. to Prop. 14, this is equal to

p^2 - \frac{1}{2} p(p-1) = \frac{1}{2} p(p+1).

Lemma 9

Assume that X is a chain with vertices 1, 2, ..., l and edges {1, 2}, {2, 3}, ..., {l - 1, l}.

(i) If one of the edges {2, 3}, {3, 4}, ..., {l - 2, l - 1} is of order $\geq 4$ , this edge is of order 4 and the graph is the following:

!Graph with 4 nodes connected in a chain

(ii) If the edge {1, 2} is of order 5, the graph is one of the following:

!Graphs with 5 nodes connected in chains

We can assume that $l > 2$ (Lemma 4). Assume that $\{i, i + 1\}$ is of order $\geq 4$ , with $1 \leq i \leq l - 1$ . Put

x = e_1 + 2e_2 + \cdots + ie_i,\ y = e_i + 2e_{l-1} + \cdots + (l - i)e_{i+1}, \text{ and } j = l - i.

By Lemma 8, $\| x \| ^2 = \frac{1}{2} i(i + 1)$ , $\| y \| ^2 = \frac{1}{2} j(j + 1)$ . On the other hand, $(x|y) = ij(e_i|e_{i+1}) = -ij \cos \frac{\pi}{m}$ with $m = 4$ or 5 (Lemma 4). Now

(x|y)^2 < \| x \| ^2 \| y \| ^2.

which gives

\frac{1}{4} ij(i + 1)(j + 1) > i^2 j^2 \cos^2 \frac{\pi}{m}

(i + 1)(j + 1) > 4ij \cos^2 \frac{\pi}{m} \geq 2ij.

This gives, first of all, $ij - i - j - 1 < 0$ , or $(i - 1)(j - 1) < 2$ . If

1 < i < l - 1,

then $1 < j < l - 1$ , so $i = j = 2$ , and so $^1$

9 > 16 \cos^2 \frac{\pi}{m}, \text{ thus } \cos^2 \frac{\pi}{m} < \cos^2 \frac{\pi}{5}

and hence $m = 4$ . This proves (i). If $i = 1$ and $m = 5$ , then $2j + 2 > 4j \frac{3+\sqrt{5}}{8}$ , or $j \frac{\sqrt{5}-1}{2} < 2, j < \sqrt{5} + 1 < 4$ , and so $l = j + 1 \leq 4$ . This proves (ii).

$^1$ The 5th roots of 1 distinct from 1 are the solutions of $z^4 + z^3 + z^2 + z + 1 = 0$ . Putting $x = \frac{1}{2}(z + \frac{1}{z})$ , this equation becomes $(2x)^2 - 2 + 2x + 1 = 0$ , or $4x^2 + 2x - 1 = 0$ , so $x = \frac{-1 \pm \sqrt{5}}{4}$ . Hence,

2 \cos^2 \frac{\pi}{5} - 1 = \cos \frac{2\pi}{5} = \frac{\sqrt{5} - 1}{4}, \quad \cos^4 \frac{\pi}{5} = \frac{3 + \sqrt{5}}{8} > \frac{5}{8} > \frac{9}{16}, \quad \cos \frac{\pi}{5} = \frac{1 + \sqrt{5}}{4}.

Lemma 10

If X has a ramification point i, the full subgraph X - {i} is the union of three chains, and if p - 1, q - 1, r - 1 are the lengths of these chains, the triple {p, q, r} is equal, up to a permutation, to one of the triples {1, 2, 2}, {1, 2, 3}, {1, 2, 4}, {1, 1, m} (for some m $\geq 1$ ).

The vertex i belongs to 3 edges (Lemma 2), and there is no other ramification point (Lemma 7), so the full subgraph X - {i} is the sum of 3 chains X_1, X_2, X_3, each of which has a terminal vertex linked to i in X. Let {i_1, i_2}, {i_2, i_3}, ..., {i_{p-1}, i_p} be the edges of X_1, {j_1, j_2}, {j_2, j_3}, ..., {j_{q-1}, j_q} those of X_2, and {k_1, k_2}, {k_2, k_3}, ..., {k_{r-1}, k_r} those of X_3, with i_1, j_1, k_1 linked to i in X. We can assume that $p \geq q \geq r \geq 1$ . Put

x = e_{i_p} + 2e_{i_{p-1}} + \cdots + pe_{i_1}

y = e_{j_q} + 2e_{j_{q-1}} + \cdots + qe_{j_1}

z = e_{k_r} + 2e_{k_{r-1}} + \cdots + re_{k_1}.

Since all the edges of X are of order 3 (Lemma 7), Lemma 8 gives $\| x \| ^2 = \frac{1}{2} p(p+1)$ , $\| y \| ^2 = \frac{1}{2} q(q+1)$ , $\| z \| ^2 = \frac{1}{2} r(r+1)$ . On the other hand, $e_i$ is orthogonal to $e_{i_2}, e_{i_3}, \ldots, e_{i_p}$ , so $(e_i | x) = p(e_i | e_{i_1}) = -\frac{1}{2} p$ ; similarly $(e_i | y) = -\frac{1}{2} q$ , $(e_i | z) = -\frac{1}{2} r$ . The unit vectors $\| x \|^{-1} x, \| y \|^{-1} y, \| z \|^{-1} z$ are mutually orthogonal, and $e_i$ does not belong to the subspace F that they generate; the square of the distance from $e_i$ to F is

1 - (e_i | \frac{x}{\| x \|})^2 - (e_i | \frac{y}{\| y \|})^2 - (e_i | \frac{z}{\| z \|})^2

= 1 - \frac{1}{2} \frac{p}{p+1} - \frac{1}{2} \frac{q}{q+1} - \frac{1}{2} \frac{r}{r+1}

= 1 - \frac{1}{2} + \frac{1}{2} \frac{1}{p+1} - \frac{1}{2} + \frac{1}{2} \frac{1}{q+1} - \frac{1}{2} + \frac{1}{2} \frac{1}{r+1}.

Since this quantity is > 0, we have

(p+1)^{-1} + (q+1)^{-1} + (r+1)^{-1} > 1.

(4)

Hence $3(r+1)^{-1} > 1$ , so $r < 2$ and finally $r = 1$ . Then (4) gives

(p+1)^{-1} + (q+1)^{-1} > \frac{1}{2}

(5)

hence $2(q+1)^{-1} > \frac{1}{2}$ , so $q \leq 2$ . Finally, if $q = 2$ , (5) gives

(p+1)^{-1} > \frac{1}{6}. \quad \text{hence } p \leq 4.

Theorem 1

The graph of any irreducible finite Coxeter system (W, S) is isomorphic to one of the following:

A_l (l \geq 1\ vertices) B_l (l \geq 2\ vertices) D_l (l \geq 4\ vertices) E_6 E_7 E_8 F_4 4 G_2 6 H_3 5 H_4 5 I_2(p) p (p = 5\ or\ p \geq 7).

No two of these graphs are isomorphic.

Indeed, let W = (m_{ij}) be the Coxeter matrix of (W, S), and let l = Card(S). If one of the m_{ij} is \geq 6, then l = 2 (Lemma 4) and the Coxeter graph of (W, S) is of type G_2 or I_2(p) with p \geq 7. Assume now that all the m_{ij} \leq 5.

a) If the m_{ij} are not all equal to 3, the graph X of (W, S) is a chain and exactly one of the m_{ij} is equal to 4 or 5 (Lemma 7). If one of the m_{ij} is equal to 5, Lemma 9 shows that we have one of the types H_3, H_4 or I_2(5). If one of the m_{ij} is equal to 4, Lemma 9 shows that we have one of the types B_l or F_4.

b) Assume now that all the m_{ij} are equal to 3. If X is a chain, the Coxeter graph is of type A_l. If not, Lemma 10 shows that it is of type E_6, E_7, E_8 or D_l.

The fact that no two of the Coxeter graphs listed are isomorphic is clear.

Conversely:

Theorem 2

The Coxeter groups defined by the Coxeter graphs A_l, B_l, ..., I_2(p) of Th. 1 are finite.

This is clear for I_2(p), the corresponding group being the dihedral group of order 2p (Chap. IV, § 1, no. 9). For H_4 the corresponding quadratic form is

\xi_1^2 + \xi_2^2 + \xi_3^2 + \xi_4^2 - \xi_1 \xi_2 - \xi_2 \xi_3 - 2(\cos \frac{\pi}{5}) \xi_3 \xi_4

= \xi_1^2 + \xi_2^2 + \xi_3^2 + \xi_4^2 - \xi_1 \xi_2 - \xi_2 \xi_3 - \frac{1 + \sqrt{5}}{2} \xi_3 \xi_4

= (\xi_2 - \frac{\xi_1 + \xi_3}{2})^2 + (\xi_4 - \frac{1 + \sqrt{5}}{4} \xi_3)^2 + \frac{3}{4} (\xi_1 - \frac{1}{3} \xi_3)^2 + \frac{7 - 3 \sqrt{5}}{24} \xi_3^2.

Since $7 - 3 \sqrt{5}$ is $> 0$ , this form is positive non-degenerate, and the corresponding Coxeter group is finite. The same holds for that corresponding to $H_3$ , since it is isomorphic to a subgroup of the preceding (Chap. IV, § 1, no. 8).

For the types $A_l, B_l, \ldots, G_2$ , we shall construct, in Nos. 5 to 13, root systems having the corresponding groups as Weyl groups. We shall see that these groups are not only finite, but crystallographic (§ 2, no. 5).

2. Dynkin graphs

By abuse of language, we shall call a normed graph a pair $(\Gamma, f)$ having the following properties:

1) $\Gamma$ is a graph (called the underlying graph of $(\Gamma, f)$ ).

2) If $E$ denotes the set of pairs $(i, j)$ such that $\{i, j\}$ is an edge of $\Gamma$ , $f$ is a map from $E$ to $\mathbf{R}$ such that $f(i, j) f(j, i) = 1$ for all $(i, j) \in E$ .

There is an obvious notion of isomorphism of normed graphs.

Let $R$ be a reduced root system in a real vector space $V$ . We are going to associate to it a normed graph $(X, f)$ , called the Dynkin graph of $R$ . The vertices of $X$ will be the elements of the set $I$ of orbits of $W(R)$ in the union of the sets $\{B\} \times B$ (where $B$ is the set of bases of $R$ ). If $N = (n_{ij})_{i,j \in I}$ (resp. $M = (m_{ij})_{i,j \in I}$ ) is the canonical Cartan matrix (resp. the Coxeter matrix) of $R$ (§ 1, no. 5, Remark 7), two vertices $i$ and $j$ of $X$ are linked if and only if $n_{ij} \neq 0$ and we then put

f(i, j) = \frac{n_{ij}}{n_{ji}}.

Since

n_{ij} = 0

implies

n_{ji} = 0

, this defines a normed graph

(X, f)

Let $(x|y)$ be a scalar product on $V$ , invariant under $W(R)$ , and let $B = (\alpha_i)_{i \in I}$ be a basis of $R$ , indexed canonically. Formulas (7) and (9) of § 1, no. 1 show that vertices $i$ and $j$ of the graph $X$ are linked if and only if

(\alpha_i|\alpha_j) \neq 0

and that

f(i, j) = \frac{(\alpha_i|\alpha_i)}{(\alpha_j|\alpha_j)}.

In view of the results of § 1, Nos. 3 and 5, the only possibilities are the following, up to interchanging $i$ and $j$ :

1) $i$ and $j$ are not linked; $n_{ij} = n_{ji} = 0; m_{ij} = 2;$ 2) $f(i, j) = f(j, i) = 1; n_{ij} = n_{ji} = -1; m_{ij} = 3;$ 3) $f(i, j) = 2, f(j, i) = 1/2; n_{ij} = -2, n_{ji} = -1; m_{ij} = 4;$ 4) $f(i, j) = 3, f(j, i) = 1/3; n_{ij} = -3, n_{ji} = -1; m_{ij} = 6.$

We see from this that the Dynkin graph $R$ determines the Cartan matrix and the Coxeter matrix of $R$ , and hence determines $R$ up to isomorphism. More precisely, the Cor. of Prop. 15 of § 1, no. 5 implies the following result:

Proposition 1

Let $R_1$ and $R_2$ be two reduced root systems in vector spaces $V_1$ and $V_2$ . Let $B_1 = (\alpha_i)_{i \in I_1}$ and $B_2 = (\alpha_i)_{i \in I_2}$ be bases of $R_1$ and $R_2$ , indexed canonically. Let $\lambda$ be an isomorphism from the Dynkin graph of $R_1$ to the Dynkin graph of $R_2$ . Then, there exists a unique isomorphism from $V_1$ to $V_2$ transforming $R_1$ into $R_2$ and $\alpha_i$ into $\alpha_{\lambda(i)}$ for all $i \in I_1$ .

It is clear that an automorphism of $R$ defines an automorphism of the Dynkin graph of $R$ , and hence a homomorphism $\varphi$ from the group $A(R)$ to the group of automorphisms of the Dynkin graph of $R$ .

Corollary

The homomorphism $\varphi$ defines by passage to the quotient an isomorphism from the group $A(R)/W(R)$ to the group of automorphisms of the Dynkin graph of $R$ .

Clearly, $\varphi(g) = \mathrm{Id}$ for all $g \in W(R)$ . On the other hand, Prop. 1 shows that there exists an isomorphism $\psi$ from the group of isomorphisms of the Dynkin graph of $R$ to the subgroup $E$ of elements of $A(R)$ leaving fixed a given basis $B$ of $R$ , such that $\varphi \circ \psi = \mathrm{Id}$ . Since $A(R)$ is the semi-direct product of $E$ and $W(R)$ (§ 1, no. 5, Prop. 16), the corollary follows.

In practice, the Dynkin graph $(X, f)$ is represented by a diagram composed of nodes and bonds in the following way. The nodes correspond to the vertices of $X$ ; two nodes corresponding to two distinct vertices $i$ and $j$ are joined by 0, 1, 2 or 3 bonds in cases 1), 2), 3) and 4) above (up to interchanging $i$ and $j$ ). Moreover, in cases 3) and 4), that is when $f(i, j) > 1$ , or when the roots $\alpha_i$ and $\alpha_j$ are not orthogonal and not of the same length, an inequality sign $>$ is placed on the double or triple bond joining the nodes corresponding to $i$ and $j$ oriented towards the node corresponding to $j$ (that is, the shortest root):

\begin{array}{cc} \circ \Longrightarrow \circ & \text{(for } f(i, j) = 2 \text{).} \\ \circ \Rrightarrow \circ & \text{(for } f(i, j) = 3 \text{).} \end{array}

It is clear that the Dynkin graph $(X, f)$ can be recovered from this diagram.

We remark that the diagram associated to the Coxeter graph of $W(R)$ can be obtained from that associated to the Dynkin graph of $R$ by keeping the nodes and the single bonds and replacing the double (resp. triple) bonds by a bond surmounted by the number 4 (resp. 6). Conversely, given the Coxeter graph of W(R), the diagram associated to the Dynkin graph of R can be recovered using the inverse of this procedure, except for the inequality signs on the double or triple bonds. Th. 1 thus gives immediately the list of possible Dynkin graphs. More precisely:

Theorem 3

If R is an irreducible reduced root system, its Dynkin graph is isomorphic to one of the graphs represented by the following diagrams:

\begin{array}{ll} A_l & \circ-\cdots-\circ \quad (l \geq 1 \text{ vertices }) \\ B_l & \circ-\cdots-\circ-\circ \quad (l \geq 2 \text{ vertices }) \\ C_l & \circ-\cdots-\circ-\circ \quad (l \geq 3 \text{ vertices }) \\ D_l & \circ-\cdots-\circ-\circ-\circ \quad (l \geq 4 \text{ vertices }) \\ E_6 & \circ-\cdots-\circ \\ E_7 & \circ-\cdots-\circ-\circ-\circ \\ E_8 & \circ-\cdots-\circ-\circ-\circ-\circ \\ F_4 & \circ-\circ-\circ-\circ \\ G_2 & \circ-\circ-\circ \end{array}

No two of these graphs are isomorphic and there are irreducible reduced root systems having each of them as their Dynkin graph (up to isomorphism).

The first assertion follows immediately from Th. 1, in view of the preceding remarks, the fact that the Coxeter groups of the graphs H_3, H_4 and I_2(p) (for p = 5 and p \geq 7) are not crystallographic, and the fact that the two possible inequalities for the double (resp. triple) bond of the Dynkin graph associated to the Coxeter graph F_4 (resp. G_2) give isomorphic Dynkin graphs. The second assertion is clear and the third will follow from the explicit construction of an irreducible reduced root system for each type, a construction that will be carried out in Nos. 5 to 13.

Remark 1

The graph A_1 reduces to a single node; we denote it also by B_1 or C_1. The graph B_2 \circ-\circ-\circ-\circ is also denoted by C_2. The graph A_3 \circ-\cdots-\circ is also denoted by D_3. Finally, D_2 denotes the graph consisting of two unconnected nodes. (These conventions are derived from the properties of the corresponding root systems, cf. nos. 5 to 8.)

Remark 2

If (X, f) is the Dynkin graph of a reduced root system R, the Dynkin graph of the inverse system can be identified with (X, f^{-1}). In other words, the diagram associated to the Dynkin graph of R^- can be obtained from that associated to the Dynkin graph of R by reversing the inequality signs. If R is irreducible, we see that R is isomorphic to R^- unless R is of type B_l or C_l, in which case R^- is of type C_l or B_l.

3. Affine Weyl group and completed Dynkin graph

Let R be an irreducible reduced root system and let (X, f) be its Dynkin graph. We are going to define another normed graph (\tilde{X}, \tilde{f}) that we shall call the completed Dynkin graph of R. The set \tilde{I} of vertices of \tilde{X} consists of the set I of vertices of X and a vertex denoted by 0, not belonging to I. To define \tilde{f}, choose a basis B = (\alpha_i)_{i \in I} of R and a scalar product \langle x | y \rangle invariant under W(R). Let \alpha_0 be the negative of the highest root for the order defined by B. Two distinct vertices i, j \in \tilde{I} are linked if and only if \langle \alpha_i | \alpha_j \rangle \neq 0 and we then put

\tilde{f}(i, j) = \frac{\langle \alpha_i | \alpha_i \rangle}{\langle \alpha_j | \alpha_j \rangle}.

It is immediate that the graph \tilde{X} and the map \tilde{f} thus defined do not depend on the choice of B or the scalar product.

If the rank l of R is equal to 1, then I = \{i\} and \alpha_0 = -\alpha_1; hence \tilde{f}(0, i) = 1. If l \geq 2, \alpha_0 is not proportional to any of the \alpha_i and \langle \alpha_0 | \alpha_i \rangle is \leq 0 (§ 1, no. 8, Prop. 25). For any pair (i, j) of distinct elements of \tilde{I}, the only possibilities are those denoted by 1), 2), 3), 4) in the preceding number (putting, for example, n_{0i} = n(\alpha_0, \alpha_i) and m_{0i} = order of s_{\alpha_0} s_{\alpha_i} for all i \in I).

In the case l \geq 2, the completed Dynkin graph is represented by a diagram with the same conventions as in the preceding number, sometimes indicating by dotted lines the bonds joining the vertex 0 to the other vertices. We remark that the inequality sign > on such a bond, if it exists, is always directed towards the vertex distinct from 0, since \alpha_0 is a longest root (§ 1, no. 8, Prop. 25). The graph (X, f) is the subgraph obtained from (\tilde{X}, \tilde{f}) by deleting the vertex 0.

The action of A(R) on (X, f) extends to an action on (\tilde{X}, \tilde{f}), leaving 0 fixed, and W(R) acts trivially on (\tilde{X}, \tilde{f}).

We retain the notations of § 2, Prop. 5 of § 2, no. 2, together with Th. 1 of Chap. V, § 3, no. 2, shows that the Coxeter graph \Sigma of the affine Weyl group W_a(R) can be obtained from (\tilde{X}, \tilde{f}) by the same rules by which the Coxeter graph of W(R) is obtained from (X, f). On the other hand, let G be the normaliser of W_a(R) (§ 2, no. 3). To any g \in G corresponds an automorphism \varphi(g) of \Sigma and \varphi(g) = \mathrm{Id} if g \in W_a(R). Conversely, given an automorphism \lambda of \Sigma there is, by Prop. 11 of Chap. V, § 4, no. 9, a unique element g = \psi(\lambda) preserving a given alcove C and such that \varphi(g) = \lambda. Since G is the semi-direct product of the subgroup G_C of elements preserving C and W_a(R) (§ 2, no. 3), we deduce that \varphi defines by passage to the quotient an isomorphism from G/W_a (or from G_C) to Aut(\Sigma). It is immediate that the composite of this isomorphism with the canonical map from A(R)/W(R) to G/W_a coincides with the homomorphism from A(R)/W(R) to Aut(\Sigma) induced by the homomorphism from A(R)/W(R) to Aut(\tilde{X}, \tilde{f}) defined above. By § 2, no. 3, the group Aut(\Sigma) is isomorphic to the semi-direct product of

A(R)/W(R) by P(R^*)/Q(R^*). and P(R^*)/Q(R^*) is isomorphic to the group $I_C = G_C \cap W'_a$ (with the notation of § 2, no. 3): the element of Aut( $\Sigma$ ) corresponding to the element $\gamma_i$ of $I_C$ transforms the vertex 0 to the vertex $i$ of $\Sigma$ .

Remark

It can be shown that the canonical map

\operatorname{Aut}(\tilde{X}, \tilde{f}) \to \operatorname{Aut}(\Sigma)

is an isomorphism.

Theorem 4

Let (W, S) be an irreducible Coxeter system with S finite. The associated quadratic form (Chap. V, § 4, no. 1) is positive and degenerate if and only if the Coxeter graph of (W, S) is isomorphic to one of the following:

\begin{array}{ll} \tilde{A}_1 & \circ \longrightarrow \infty \\ \tilde{A}_l \quad (l \geq 2) & \circ \longrightarrow \cdots \longrightarrow \circ \quad \text{(circuit with } l + 1 \text{ vertices)} \\ \tilde{B}_2 & \circ \longrightarrow 4 \longrightarrow \circ \\ \tilde{B}_l \quad (l \geq 3) & \circ \longrightarrow \cdots \longrightarrow \circ \longrightarrow 4 \quad \text{(l + 1 vertices)} \\ \tilde{C}_l \quad (l \geq 3) & \circ \longrightarrow 4 \longrightarrow \cdots \longrightarrow 4 \quad \text{(l + 1 vertices)} \\ \tilde{D}_l \quad (l \geq 4) & \circ \longrightarrow \cdots \longrightarrow \circ \quad \text{(l + 1 vertices)} \\ \tilde{E}_6 & \circ \longrightarrow \circ \longrightarrow \circ \longrightarrow \circ \\ \tilde{E}_7 & \circ \longrightarrow \circ \longrightarrow \circ \longrightarrow \circ \longrightarrow \circ \\ \tilde{E}_8 & \circ \longrightarrow \circ \longrightarrow \circ \longrightarrow \circ \longrightarrow \circ \longrightarrow \circ \\ \tilde{F}_4 & \circ \longrightarrow \circ \longrightarrow 4 \longrightarrow \circ \\ \tilde{G}_2 & \circ \longrightarrow 6 \longrightarrow \circ \end{array}

No two of these Coxeter graphs are isomorphic.

By Chap. V, § 4, no. 9 and Prop. 8 of § 2, no. 5, the Coxeter systems whose quadratic form is positive and degenerate are those which correspond to the affine Weyl groups of irreducible reduced root systems. The theorem therefore follows from the determination of the completed Dynkin graphs made in nos. 5 to 13 below.

4. Preliminaries to the construction of root systems

Let V be a real vector space of dimension $l \geq 1$ equipped with a scalar product $(x|y)$ , L a discrete subgroup of V, $\Lambda$ a finite set of numbers > 0, and R the set of $\alpha \in L$ such that $(\alpha|\alpha) \in \Lambda$ . Assume that R generates V and that, for any pair $(\alpha, \beta)$ of points of R, the number $2 \frac{(\alpha|\beta)}{(\alpha|\alpha)}$ is an integer. Then, R is a root system in V. Indeed, R clearly satisfies (RS_I). Let $\alpha \in R$ ; let $s_{\alpha}$ be the orthogonal reflection $x \mapsto x - 2 \frac{(x|\alpha)}{(\alpha|\alpha)} \alpha$ : then, if $\beta \in R$ , we have $2 \frac{(\beta|\alpha)}{(\alpha|\alpha)} \in \mathbf{Z}$ , so $s_{\alpha}(\beta) \in L$ , and moreover $\| s_{\alpha}(\beta) \| = \| \beta \|$ , so $s_{\alpha}(\beta) \in R$ ; thus, R satisfies (RS_{II}) and (RS_{III}), and is reduced if $\Lambda$ does not contain two numbers of the form $\lambda$ and $4\lambda$ .

Now let V be a subspace of $E = \mathbf{R}^n$ . Let $(\varepsilon_1, \ldots, \varepsilon_n)$ be the canonical basis of E; we equip E with the scalar product for which this basis is orthonormal and identify $E^*$ with E (resp. $V^*$ with V) by means of this scalar product. We define subgroups $L_0, L_1, L_2, L_3$ of E as follows:

1) $L_0$ is the $\mathbf{Z}$ -module with basis $(\varepsilon_i)$ . We have $(\alpha|\beta) \in \mathbf{Z}$ for all $\alpha, \beta \in L_0$ . The vectors $\alpha \in L_0$ for which $(\alpha|\alpha) = 1$ are the $\pm \varepsilon_i$ ( $1 \leq i \leq n$ ); those for which $(\alpha|\alpha) = 2$ are the $\pm \varepsilon_i \pm \varepsilon_j$ for $i < j$ (the two signs $\pm$ , in $\pm \varepsilon_i \pm \varepsilon_j$ , are chosen independently of each other; we adopt an analogous convention throughout the remainder of this section).

2) $L_1$ is the $\mathbf{Z}$ -submodule of $L_0$ consisting of the $x = \sum_{i=1}^n \xi_i \varepsilon_i \in L_0$ such that $\sum_{i=1}^n \xi_i$ is even; since $\xi_i$ and $\xi_i^2$ have the same parity, this is equivalent to requiring that $(x|x)$ is even. Let $L_1'$ be the submodule of $L_1$ generated by the $\varepsilon_i \pm \varepsilon_j$ ; we have $\sum_{i=1}^n \xi_i \varepsilon_i \equiv (\sum_{i=1}^n \xi_i) \varepsilon_n \pmod{L_1'}$ , and since $2\varepsilon_n = (\varepsilon_1 + \varepsilon_n) - (\varepsilon_1 - \varepsilon_n) \in L_1'$ , it follows that $L_1' = L_1$ . Since $L_0$ is generated by $L_1$ and $\varepsilon_1$ , $L_0/L_1$ is isomorphic to $\mathbf{Z}/2\mathbf{Z}$ .

3) $L_2 = L_0 + \mathbf{Z}(\frac{1}{2} \sum_{i=1}^n \varepsilon_i)$ . It is clear that an element $x = \sum_{i=1}^n \xi_i \varepsilon_i$ of V is in $L_2$ if and only if

2\xi_i \in \mathbf{Z}, \quad \xi_i - \xi_j \in \mathbf{Z} \text{ for all } i \text{ and } j.

Since

(\varepsilon_k | \frac{1}{2} \sum_{i=1}^n \varepsilon_i ) = \frac{1}{2}

for all

k

, and since

\| \frac{1}{2} \sum_{i=1}^n \varepsilon_i \| ^2 = \frac{n}{4}

, we have

(\alpha|\beta) \in \frac{1}{2}\mathbf{Z}

for

\alpha, \beta \in L_2

n

is even. The group

L_2/L_0

is isomorphic to

\mathbf{Z}/2\mathbf{Z}

4) $L_3 = L_1 + \mathbf{Z} (\frac{1}{2} \sum_{i=1}^n \varepsilon_i )$ . If $n$ is a multiple of 4, $L_3$ is the set of $\sum_{i=1}^n \xi_i \varepsilon_i$ satisfying (6) and the condition $\sum_{i=1}^n \xi_i \in 2\mathbf{Z}$ : in this case, $(\alpha|\beta) \in \mathbf{Z}$ for all $\alpha, \beta \in L_3$ .

It is immediate that the subgroup of E associated to $L_0$ (resp. $L_1, L_2$ ) is $L_0$ (resp. $L_2, L_1$ ). The subgroup of E associated to $L_3$ is the set of

x := \sum_{i=1}^n \xi_i \varepsilon_i \in L_2

such that $(x| \frac{1}{2} \sum_{i=1}^n \varepsilon_i ) \in \mathbf{Z}$ , that is, such that $\sum_{i=1}^n \xi_i \in 2\mathbf{Z}$ : if $n \equiv 0 \pmod{4}$ , this associated subgroup is therefore $L_3$ .

The abelian group $L_2/L_1$ is of order 4, and hence is isomorphic to $\mathbf{Z}/4\mathbf{Z}$ or $\mathbf{Z}/2\mathbf{Z} \times \mathbf{Z}/2\mathbf{Z}$ (Algebra, Chap. VII, § 4, no. 6, Th. 3). If $n$ is odd,

p(\frac{1}{2} \sum_{i=1}^n \varepsilon_i ) \in L_1 \Leftrightarrow p \equiv 0 \pmod{4}

so $L_2/L_1$ is cyclic of order 4. If $n$ is even,

p(\frac{1}{2} \sum_{i=1}^n \varepsilon_i ) \in L_1 \Leftrightarrow p \equiv 0 \pmod{2}

so $L_2/L_1$ , which contains two distinct elements of order 2, is isomorphic to $\mathbf{Z}/2\mathbf{Z} \times \mathbf{Z}/2\mathbf{Z}$ .

We shall use this notation throughout the next nine nos. and in the plates. For each type of Dynkin graph in Th. 3, we shall describe explicitly:

(I) A root system R and the number of roots. (II) A basis B of R, and the corresponding positive roots. The basis B will be indexed by the integers 1, ..., l. (III) The Coxeter number h (§ 1, no. 11). (IV) The highest root $\tilde{\alpha}$ (for the order defined by B) and the completed Dynkin graph (no. 3). We shall indicate next to each vertex the corresponding root of B. (V) The inverse system $R^-$ , the canonical bilinear form and the constant $\gamma(R)$ (§ 1, no. 12). (VI) The fundamental weights relative to B (§ 1, no. 10). (VII) The sum of the positive roots. (VIII) The groups P(R), Q(R), P(R)/Q(R) and the connection index (§ 1, no. 9). (IX) The exponents of W(R) (Chap. V, § 6, no. 2, Def. 2). In the cases A_l, B_l, C_l and D_l we determine the symmetric invariants. (X) The order of W(R) (and in some cases its structure).

(XI) The group $A(R)/W(R)$ , its action on the Dynkin graph, and the element $w_0$ of $W(R)$ that transforms $B$ to $-B$ .

(XII) The action of $P(R^-)/Q(R^-)$ on the completed Dynkin graph and the action of $A(R)/W(R)$ on $P(R^-)/Q(R^-)$ .

For each Dynkin graph in Th. 3, these data will be collected in Plates I to IX, and ordered in a uniform way as above. We also give:

(XIII) The Cartan matrix, from which the Dynkin graph is derived as described in no. 2.

5. Systems of type $B_l$ ( $l \geq 2$ )

(I) We consider the group $L_0$ in $V = \mathbf{R}^l$ (no. 4). Let $R$ be the set of $\alpha \in L_0$ such that $(\alpha|\alpha) = 1$ or $(\alpha|\alpha) = 2$ , in other words the set of vectors $\pm \varepsilon_i$ ( $1 \leq i \leq l$ ) and $\pm \varepsilon_i \pm \varepsilon_j$ ( $1 \leq i < j \leq l$ ). It is clear that $R$ generates $V$ and that $2(\alpha|\beta)/(\alpha|\alpha) \in \mathbf{Z}$ for all $\alpha, \beta \in R$ . Thus $R$ is a reduced root system in $V$ (no. 4). The number of roots is $n = 2l + 4^{l(l-1)/2} = 2l^2$ .

(II) Put

\alpha_1 = \varepsilon_1 - \varepsilon_2,\ \alpha_2 = \varepsilon_2 - \varepsilon_3,\ldots,\alpha_{l-1} = \varepsilon_{l-1} - \varepsilon_1,\ \alpha_l = \varepsilon_l.

Then

\varepsilon_i = \alpha_i + \alpha_{i+1} + \cdots + \alpha_l \quad (1 \leq i \leq l)

\varepsilon_i + \varepsilon_j = (\alpha_i + \alpha_{i+1} + \cdots + \alpha_l) + (\alpha_j + \alpha_{j+1} + \cdots + \alpha_l) \quad (1 \leq i < j \leq l)

\varepsilon_i - \varepsilon_j = \alpha_i + \alpha_{i+1} + \cdots + \alpha_{j-1} \quad (1 \leq i < j \leq l).

Thus $(\alpha_1, \alpha_2, \ldots, \alpha_l)$ is a basis of $R$ (§ 1, no. 7, Cor. 3 of Prop. 20). Moreover, $\| \alpha_i \|^2 = 2$ for $i < l$ , $\| \alpha_l \|^2 = 1$ , $(\alpha_i|\alpha_{i+1}) = -1$ for $1 \leq i \leq l-1$ , $(\alpha_i|\alpha_j) = 0$ for $j > i + 1$ : the Dynkin graph of $R$ is thus of type $B_l$ , which shows that $R$ is irreducible. The positive roots are the $\varepsilon_i$ and the $\varepsilon_i \pm \varepsilon_j$ ( $i < j$ ).

(III) By Th. 1 (ii) of Chap. V, § 6, no. 2,

h = n/l = 2l.

(IV) Let $\tilde{\alpha} = \varepsilon_1 + \varepsilon_2 = \alpha_1 + 2\alpha_2 + 2\alpha_3 + \cdots + 2\alpha_l$ , which is a root. The sum of its coordinates relative to the basis $(\alpha_i)$ is $2l - 1 = h - 1$ . In view of Prop. 31 of § 1, no. 11, $\tilde{\alpha}$ is the highest root of $R$ . We have $(\tilde{\alpha}|\alpha_i) = 0$ for $i \neq 2$ and $(\tilde{\alpha}|\alpha_2) = 1$ . Since $\alpha_2$ is of length 1 (resp. $\sqrt{2}$ ) when $l = 2$ (resp. $l \geq 3$ ), the completed Dynkin graph of $R$ is as follows:

for $l = 2$

\begin{array}{c} \alpha_2 \\ \alpha_1 \end{array}

for $l \geq 3$

\begin{array}{cccccc} \alpha_1 & & & & & \\ & \alpha_2 & \alpha_3 & \cdots & \alpha_{l-1} & \alpha_l \end{array}

(V) The formula $\alpha^- = \frac{2\alpha}{(\alpha|\alpha)}$ gives for $R^-$ the set of vectors $\pm 2\varepsilon_i$ ( $1 \leq i \leq l$ ), $\pm \varepsilon_i \pm \varepsilon_j$ ( $1 \leq i < j \leq l$ ). The Dynkin graph of $R^-$ is obtained from that of $R$ by the procedure explained in no. 2, and we see that $R^-$ is of type $C_l$ .

There are $4l - 2$ roots not orthogonal to $\beta = \varepsilon_1$ , namely $\pm \varepsilon_1$ and $\pm \varepsilon_1 \pm \varepsilon_j$ for $2 \leq j \leq l$ ; for each such root $\alpha$ , $n(\alpha, \beta) = \pm 2$ . Formula (17) of § 1, no. 12 shows that, for $\Phi_R$ , the square of the length of $\beta$ is $(4l - 2)^{-1}$ : hence $\Phi_R(x, y) = (x|y)/(4l - 2)$ . Apply formula (18) of § 1, no. 12 with $x = y = \beta$ . This gives

2 + \frac{1}{4}(4l - 4) = \gamma(R) \frac{1}{4l - 2}

and so

\gamma(R) = (l + 1)(4l - 2)

(VI) The fundamental weights $\omega_i$ ( $1 \leq i \leq l$ ) such that $(\omega_i|\alpha_j) = \delta_{ij}$ are easily calculated, and we find that

\begin{align*} \omega_i &= \varepsilon_1 + \varepsilon_2 + \cdots + \varepsilon_i \\ &= \alpha_1 + 2\alpha_2 + \cdots + (i - 1)\alpha_{i-1} + i(\alpha_1 + \alpha_{i+1} + \cdots + \alpha_l) \quad (i < l) \\ \omega_l &= \frac{1}{2}(\varepsilon_1 + \varepsilon_2 + \cdots + \varepsilon_l) = \frac{1}{2}(\alpha_1 + 2\alpha_2 + \cdots + l\alpha_l). \end{align*}

(VII) The sum of the positive roots is

\begin{align*} 2\rho &= (2l - 1)\varepsilon_1 + (2l - 3)\varepsilon_2 + \cdots + 3\varepsilon_{l-1} + \varepsilon_l \\ &= (2l - 1)\alpha_1 + 2(2l - 2)\alpha_2 + \cdots + i(2l - i)\alpha_i + \cdots + l^2\alpha_l. \end{align*}

(VIII) We have $Q(R) = L_0$ (no. 4), and $P(R)$ is generated by $Q(R)$ and $\omega_l$ , hence is equal to $L_2$ (no. 4). Hence, $P(R)/Q(R)$ is isomorphic to $\mathbf{Z}/2\mathbf{Z}$ , and the connection index is equal to 2.

(IX) and (X) In $\mathbf{R}^l$ , the orthogonal reflection $s_{\varepsilon_i, -\varepsilon_j}$ ( $i \neq j$ ) interchanges $\varepsilon_i$ and $\varepsilon_j$ and leaves $\varepsilon_k$ invariant when the index $k$ is distinct from $i$ and $j$ . The $s_{\varepsilon_i, -\varepsilon_i}$ generate a group $G_1$ isomorphic to the symmetric group $S_l$ . The orthogonal reflection $s_{\varepsilon_i}$ transforms $\varepsilon_i$ to $-\varepsilon_i$ and leaves invariant the $\varepsilon_k$ when the index $k$ is distinct from $i$ . The $s_{\varepsilon_i}$ generate a group $G_2$ isomorphic to $(\mathbf{Z}/2\mathbf{Z})^l$ . The Weyl group $W(R)$ is generated by $G_1$ and $G_2$ , and $G_2$ is normal in $W(R)$ , so $W(R)$ is isomorphic to the semi-direct product of $G_l$ by $(\mathbf{Z}/2\mathbf{Z})^l$ . Its order is therefore $2^l.l!$ .

The symmetric algebra $S(\mathbf{R}^l)$ can be identified canonically with the algebra of polynomial functions $P(\xi_1, \ldots, \xi_l)$ on $\mathbf{R}^l$ . For such a polynomial to be invariant under $W(R)$ , it is first of all necessary that

P(\xi_1, \xi_2, \ldots, \xi_l) = P(\pm \xi_1, \pm \xi_2, \ldots, \pm \xi_l)

for all choices of the signs on the right-hand side, so that

P(\xi_1, \ldots, \xi_l) = Q(\xi_1^2, \ldots, \xi_l^2)

where Q is a polynomial, and further that Q is a symmetric polynomial; and these conditions are sufficient. Consequently (Algebra. Chap. V, App. I), $S(\mathbf{R}^l)^{W(R)}$ is the algebra generated by the l polynomial functions

t_i = \sum_{\tau \in S_l} \xi_{\tau(1)}^2 \xi_{\tau(2)}^2 \cdots \xi_{\tau(l)}^2 \quad (1 \leq i \leq l).

Moreover, the transcendence degree over $\mathbf{R}$ of the field of fractions of $S(\mathbf{R}^l)^{W(R)}$ is l, so the $t_i$ are algebraically independent. Since the $t_i$ have degree 2, 4, ..., 2l, we conclude that the exponents of W(R) (Chap. V. § 6, no. 3, Prop. 3) are:

1, 3, 5, \ldots, 2l - 1.

(XI) The only automorphism of the Dynkin graph is the identity element. Thus, $A(R) = W(R)$ and $-1 \in W(R)$ . Since $-1$ transforms B to $-B$ , we conclude that $w_0 = -1$ .

(XII) The group $P(R^-)/Q(R^-)$ is dual to $P(R)/Q(R)$ , and hence is isomorphic to $\mathbf{Z}/2\mathbf{Z}$ . Its non-trivial element permutes the vertices corresponding to $\alpha_0$ and $\alpha_1$ and leaves the others fixed.

6. Systems of type $C_l$ ( $l \geq 2$ )

(I) The existence of root systems of type $C_l$ has been proved in no. 5, since we have seen that the inverse system of a system of type $B_l$ is of type $C_l$ . A root system of type $C_l$ is thus obtained by taking in $\mathbf{R}'$ the vectors $\pm 2\varepsilon_i$ ( $1 \leq i \leq l$ ), and $\pm \varepsilon_i \pm \varepsilon_j$ ( $1 \leq i < j \leq l$ ). The number of roots is $2l^2$ .

(II) A basis of R can be obtained by taking the image under the map $\alpha \mapsto \frac{2\alpha}{(\alpha|\alpha)}$ of the basis of the system considered in no. 5. We obtain:

\alpha_1 = \varepsilon_1 - \varepsilon_2, \quad \alpha_2 = \varepsilon_2 - \varepsilon_3, \ldots, \quad \alpha_{l-1} = \varepsilon_{l-1} - \varepsilon_l, \quad \alpha_l = 2\varepsilon_l.

The positive roots are the $2\varepsilon_i$ and the $\varepsilon_i \pm \varepsilon_j$ ( $i < j$ ).

(III) The Coxeter number is the same as for the inverse system: $h = 2l$ .

(IV) Let $\tilde{\alpha} = 2\varepsilon_1 = 2\alpha_1 + 2\alpha_2 + \cdots + 2\alpha_{l-1} + \alpha_l$ , which is a root. The sum of its coordinates relative to $(\alpha_i)$ is $2l - 1 = h - 1$ . Hence, $\tilde{\alpha}$ is the highest root. We have $(\tilde{\alpha}|\alpha_i) = 0$ for $i \neq l$ , $(\tilde{\alpha}|\alpha_l) = 2$ . Thus, the completed Dynkin graph is

!Dynkin graph for C_l

(V) We have already determined $R^-$ , which is of type $B_l$ . By formula (19) of § 1, no. 12 and by no. 5 (V), the square of the length of $2\varepsilon_i$ for $\Phi_R$ is

((l+1)(4l-2))^{-1} ((4l-2)^{-1})^{-1} = (l+1)^{-1};

thus, $\Phi_R(x, y) = (x|y)/4(l+1)$ .

We have $\gamma(R) = \gamma(R^-) = (l+1)(4l-2)$ .

(VI) The fundamental weights are easily found:

\omega_i = \varepsilon_1 + \varepsilon_2 + \cdots + \varepsilon_i

= \alpha_1 + 2\alpha_2 + \cdots + (i-1)\alpha_{i-1} + i(\alpha_i + \alpha_{i+1} + \cdots + \frac{1}{2}\alpha_l) \quad (i \leq l).

(VII) The sum of the positive roots is

2\rho = 2l\varepsilon_1 + (2l-2)\varepsilon_2 + \cdots + 4\varepsilon_{l-1} + 2\varepsilon_l

= 2l\alpha_1 + 2(2l-1)\alpha_3 + \cdots + i(2l-i+1)\alpha_i + \cdots

\cdots + (l-1)(l+2)\alpha_{l-1} + \frac{1}{2}l(l+1)\alpha_l.

(VIII) By no. 4 and no. 5 (VIII), $Q(R) = L_1, P(R) = L_0; P(R)/Q(R)$ is isomorphic to $\mathbf{Z}/2\mathbf{Z}$ , and the connection index is 2.

(IX) and (X) These data depend only on $W(R)$ , and so are the same as for type $B_l$ .

(XI) The same argument as in no. 5 shows that $A(R) = W(R)$ and $w_0 = -1$ .

(XII) The single non-identity element of $P(R^-)/Q(R^-)$ defines the unique non-trivial automorphism of the completed Dynkin graph: it interchanges the vertices corresponding to $\alpha_j$ and $\alpha_{l-j}$ for $0 \leq j \leq l$ .

7. Systems of type $A_l$ ( $l \geq 1$ )

(I) and (II) Let $V$ be the hyperplane in $E = \mathbf{R}^{l+1}$ with equation $\sum_{i=1}^{l+1} \xi_i = 0$ . Replacing $l$ by $l+1$ in no. 5, we obtain a system $R'$ of type $B_{l+1}$ in $E$ with basis

\alpha_1 = \varepsilon_1 - \varepsilon_2,\ \alpha_2 = \varepsilon_2 - \varepsilon_3, \ldots, \alpha_l = \varepsilon_l - \varepsilon_{l+1},\ \alpha_{l+1} = \varepsilon_{l+1}.

Since

\alpha_1, \ldots, \alpha_l

generate

V

R = R' \cap V

is a root system in

V

with basis

(\alpha_1, \ldots, \alpha_l)

(§ 1, no. 7, Cor. 4 of Prop. 20). By the calculation of the scalar products in no. 5, it is immediate that

R

is of type

A_l

. The elements of

R

are the

\varepsilon_i - \varepsilon_j

(

i \neq j, 1 \leq i \leq l+1, 1 \leq j \leq l+1

). There are

n = l(l+1)

roots. The positive roots are the

\varepsilon_i - \varepsilon_j

where

i < j

(III) We have $h = n/l = l+1$ .

(IV) Let $\tilde{\alpha} = \varepsilon_1 - \varepsilon_{l+1} = \alpha_1 + \alpha_2 + \cdots + \alpha_l$ , which is a root. The sum of its coordinates relative to $(\alpha_i)$ is $l = h-1$ . Hence $\tilde{\alpha}$ is the highest root. For $l = 1$ , $\tilde{\alpha} = \alpha_1$ so $(\tilde{\alpha}|\alpha_1) = 2$ ; the Coxeter graph of the group $W_a(R)$ is

\begin{array}{c} \infty \\ \alpha_1 \end{array}

For $l \geq 2$ . $(\tilde{\alpha}|\alpha_i) = 0$ for $0 < i < l$ and $(\tilde{\alpha}|\alpha_1) = (\tilde{\alpha}|\alpha_l) = 1$ . Hence the completed Dynkin graph is:

!Dynkin graph

(V) Identifying V with its dual using the scalar product, we have $\tilde{\alpha} = \frac{2\alpha}{(\alpha|\alpha)} = \alpha$ for all $\alpha \in \mathbf{R}$ , so $R^* = R$ .

For the form $\Phi_R$ , the length of the roots is $h^{-1/2} = (l+1)^{-1/2}$ (§ 1, no. 12); so $\Phi_r(x, y) = (x|y)/2(l+1)$ .

We have $\gamma(R) = (l+1)^2$ (§ 1, no. 12, formula (20)).

(VI) Let $(\omega_i)_{1 \leq i \leq l}$ be the family of fundamental weights. Put

\omega_i = \sum_{j=1}^{l+1} \xi_{ij} \varepsilon_j. \quad \text{with } \xi_{ij} \in \mathbf{R}.

The conditions $(\omega_i|\alpha_j) = \delta_{ij}$ and $\omega_i \in V$ give

\xi_{ii} - \xi_{i,i+1} = 1, \quad \xi_{ij} - \xi_{i,j+1} = 0 \text{ for } j \neq i. \quad \sum_{j=1}^{l+1} \xi_{ij} = 0,

which easily lead to

\begin{align*} \omega_i &= \varepsilon_1 + \cdots + \varepsilon_i - \frac{i}{l+1}(\varepsilon_1 + \cdots + \varepsilon_{l+1}) \\ &= \frac{1}{l+1}((l-i+1)(\alpha_1 + 2\alpha_2 + \cdots + (i-1)\alpha_{i-1}) \\ &\qquad + i((l-i+1)\alpha_i + (l-i)\alpha_{i+1} + \cdots + \alpha_l)). \end{align*}

(VII) The sum of the positive roots is

\begin{align*} 2\rho &= l\varepsilon_1 + (l-2)\varepsilon_2 + (l-4)\varepsilon_3 + \cdots - (l-2)\varepsilon_l - l\varepsilon_{l+1} \\ &= l\alpha_1 + 2(l-1)\alpha_2 + \cdots + i(l-i+1)\alpha_i + \cdots + l\alpha_l. \end{align*}

(VIII) Introduce in $E = \mathbf{R}^{l+1}$ the subgroup $L_0$ of no. 4. Let $p$ be the orthogonal projection of $E$ onto $V$ . By § 1, no. 10, Prop. 28, we have

Q(R) = Q(R') \cap V \not\subset L_0 \cap V, \quad \text{and} \quad P(R) = p(P(R'));

in view of the fact that the last fundamental weight of $R'$ is orthogonal to $V$ , we have $P(R) = p(Q(R')) = p(L_0)$ . Thus, $P(R)$ is the group generated by the $\varepsilon_i - \varepsilon_j$ and by $p(\varepsilon_1) = \varepsilon_1 - (l+1)^{-1} \sum_{i=1}^{l+1} \varepsilon_i$ , so

P(R) = Q(R) + \mathbf{Z}(\varepsilon_1 - (l+1)^{-1} \sum_{i=1}^{l+1} \varepsilon_i).

Now $l + 1$ is the smallest integer $m > 0$ such that $mp(\varepsilon_1) \in Q(R)$ . Thus $P(R)/Q(R)$ is isomorphic to $\mathbf{Z}/(l+1)\mathbf{Z}$ and the connection index is $l + 1$ .

(IX) and (X) For any automorphism $g$ of $V$ , let $\varphi(g)$ be the automorphism of $E$ that extends $g$ and leaves $\varepsilon_1 + \varepsilon_2 + \cdots + \varepsilon_l$ invariant. If $g$ is the orthogonal reflection $s_{\varepsilon_i - \varepsilon_j}|V$ , $\varphi(g)$ is equal to $s_{\varepsilon_i - \varepsilon_j}$ , which interchanges $\varepsilon_i$ and $\varepsilon_j$ and leaves fixed the $\varepsilon_k$ with $k$ distinct from $i$ and $j$ . Let

X = \{\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_{l+1}\}.

Then $g \mapsto \varphi(g)|X$ is an isomorphism from $W(R)$ to the symmetric group of $X$ . Thus, $W(R)$ is isomorphic to the symmetric group $S_{l+1}$ , and so is of order $(l+1)!$ .

The symmetric algebra $S(E)$ can be identified canonically with the algebra of polynomial functions $P(\xi_1, \xi_2, \ldots, \xi_{l+1})$ on $E$ . Let $G = \varphi(W(R))$ . By the preceding paragraph, the set $S(E)^G$ of elements of $S(E)$ invariant under $G$ is the set of symmetric polynomials (Algebra, Chap. V, App. I), and consequently (\emph{ibid.}) is the algebra generated by the functions

s'_i = \sum_{\tau \in S_{l+1}} \xi_{\tau(1)} \xi_{\tau(2)} \cdots \xi_{\tau(i)} \quad (1 \leq i \leq l+1).

The algebra $S(V)$ can be identified with the restrictions to $V$ of the polynomial functions on $E$ . If $P \in S(E)^G$ , the restriction of $P$ to $V$ is clearly invariant under $W(R)$ . Conversely, if $Q \in S(V)^{W(R)}$ , there exists $P \in S(E)$ extending $Q$ ; replacing $P$ by $((l+1)!)^{-1} \sum_{g \in G} g(P)$ , which has the same restriction as $P$ to $V$ , we can assume that $P \in S(E)^G$ . Thus, $S(V)^{W(R)}$ is generated by the $s_i = s'_i|V$ . Now $s_1 = 0$ . Moreover, the transcendence degree over $\mathbf{R}$ of the field of fractions of $S(V)^{W(R)}$ is $l$ , so the $s_i$ ( $2 \leq i \leq l+1$ ) are algebraically independent. Since the $s_i$ are of degrees $2, 3, \ldots, l+1$ , the exponents of $W(R)$ are

1, 2, 3, \ldots, l.

(XI) For $l = 1$ , $A(R) = W(R) = \mathbf{Z}/2\mathbf{Z}$ and $w_0 = -1$ .

For $l \geq 2$ , let $\varepsilon \in A(R)$ be the automorphism that transforms $\alpha_i$ to $\alpha_{l+1-i}$ . It is clear that the automorphism induced by $\varepsilon$ is the unique non-trivial automorphism of the Dynkin graph. The group $A(R)/W(R)$ is isomorphic to $\mathbf{Z}/2\mathbf{Z}$ . Since $-1$ is an element of $A(R)$ which does not belong to $W(R)$ by (IX) and (X), we see that $A(R)$ is isomorphic to $W(R) \times \mathbf{Z}/2\mathbf{Z}$ . We have $w_0 = -\varepsilon$ .

(XII) The group $P(R^-)/Q(R^-)$ is cyclic of order $l+1$ and acts on the completed Dynkin graph by circular permutations. If $l \geq 2$ , the unique non-identity element of $A(R)/W(R)$ acts on $P(R^-)/Q(R^-)$ by the automorphism $x \mapsto -x$ .

8. Systems of type $D_l$ ( $l \geq 3$ )

(I) Consider in $V = \mathbf{R}'$ the group $L_0$ (no. 4). The set $R$ of $\alpha \in L_0$ such that $(\alpha|\alpha) = 2$ consists of the vectors $\pm \varepsilon_i \pm \varepsilon_j$ ( $1 \leq i < j \leq l$ ). It is clear that $R$ generates $V$ and that $2(\alpha|\beta)/(\alpha|\alpha) \in \mathbf{Z}$ for all $\alpha, \beta \in R$ . Thus $R$ is a reduced root system in $V$ (no. 4). The number of roots is $n = 2l(l-1)$ .

(II) Put

\alpha_1 = \varepsilon_1 - \varepsilon_2,\ \alpha_2 = \varepsilon_2 - \varepsilon_3 \ldots \ldots \alpha_{l-1} = \varepsilon_{l-1} - \varepsilon_l,\ \alpha_l = \varepsilon_{l-1} + \varepsilon_l.

The following formulas are immediate:

\varepsilon_i - \varepsilon_j = \alpha_i + \alpha_{i+1} + \cdots + \alpha_{j-1}\quad (i < j)

\varepsilon_i + \varepsilon_j = \alpha_i + \alpha_{i+1} + \cdots + \alpha_{j-1} + 2\alpha_j + 2\alpha_{j+1} + \cdots

\cdots + 2\alpha_{l-2} + \alpha_{l-1} + \alpha_l\quad (i < j \leq l-2)

\varepsilon_i + \varepsilon_{l-1} = \alpha_i + \alpha_{i+1} + \cdots + \alpha_l\quad (i < l-1)

\varepsilon_i + \varepsilon_l = \alpha_i + \alpha_{i+1} + \cdots + \alpha_{l-2} + \alpha_l\quad (i < l-1)

\varepsilon_{l-1} + \varepsilon_l = \alpha_l.

so $(\alpha_1, \ldots, \alpha_l)$ is a basis if $R$ (§ 1, no. 2, Cor. 3 of Prop. 20). Further, $\| \alpha_i \| ^2 = 2$ for all $i$ . $(\alpha_i|\alpha_j) = 0$ for $i + 1 < j$ except for $i = l-2, j = l$ in which case $(\alpha_{l-2}|\alpha_l) = -1$ , $(\alpha_i|\alpha_{i+1}) = -1$ for $i \leq l-2$ , and finally $(\alpha_{l-1}|\alpha_l) = -1$ ; the Dynkin graph of $R$ is thus of type $D_l$ . The positive roots are the $\varepsilon_i \pm \varepsilon_j$ for $i < j$ .

(III) We have $h = n/l = 2(l-1)$ .

(IV) Let $\tilde{\alpha} = \varepsilon_1 + \varepsilon_2 = \alpha_1 + 2\alpha_2 + \cdots + 2\alpha_{l-2} + \alpha_{l-1} + \alpha_l$ , which is a root. The sum of its coordinates relative to $(\alpha_i)$ is

2l-3 = h-1.

Hence $\tilde{\alpha}$ is the highest root.

If $l = 3$ , we have

(\tilde{\alpha}|\alpha_1) = 0,\quad (\tilde{\alpha}|\alpha_2) = (\tilde{\alpha}|\alpha_3) = 1.

If $l \geq 4$ , we have $(\tilde{\alpha}|\alpha_i) = 0$ for $i \neq 2$ and $(\tilde{\alpha}|\alpha_2) = 1$ . Hence the completed Dynkin graph is:

!Dynkin graph for D_l

(V) Since $(\alpha|\alpha) = 2$ for all $\alpha \notin R$ , we have $R^- = R$ . The length of the roots for $\Phi_{L'}$ is $h^{-1/2} = (2l-2)^{-1/2}$ . Hence

\Phi_R(x, y) = (x|y)/(4l-4)\quad \text{and}\quad \gamma(R) = 4(l-1)^2.

(VI) A calculation analogous to that in no. 7 gives the fundamental weights:

\omega_i = \varepsilon_1 + \varepsilon_2 + \cdots + \varepsilon_i \\ = \alpha_1 + 2\alpha_2 + \cdots + (i-1)\alpha_{i-1} + i(\alpha_i + \alpha_{i+1} + \cdots + \alpha_{l-2}) \\ \quad + \frac{1}{2}i(\alpha_{l-1} + \alpha_l)

for $i < l - 1$ ,

\omega_{l-1} = \frac{1}{2}(\varepsilon_1 + \varepsilon_2 + \cdots + \varepsilon_{l-2} + \varepsilon_{l-1} - \varepsilon_l) \\ = \frac{1}{2}(\alpha_1 + 2\alpha_2 + \cdots + (l-2)\alpha_{l-2} + \frac{1}{2}l\alpha_{l-1} + \frac{1}{2}(l-2)\alpha_l),

\omega_l = \frac{1}{2}(\varepsilon_1 + \varepsilon_2 + \cdots + \varepsilon_{l-2} + \varepsilon_{l-1} + \varepsilon_l) \\ = \frac{1}{2}(\alpha_1 + 2\alpha_2 + \cdots + (l-2)\alpha_{l-2} + \frac{1}{2}(l-2)\alpha_{l-1} + \frac{1}{2}l\alpha_l).

(VII) The sum of the positive roots is

2\rho = 2(l-1)\varepsilon_1 + 2(l-2)\varepsilon_2 + \cdots + 2\varepsilon_{l-1} \\ = \sum_{i=1}^{l-2} 2(i(l-\frac{i(i+1)}{2}))\alpha_i + \frac{l(l-1)}{2}(\alpha_{l-1} + \alpha_l).

(VIII) The $\pm \varepsilon_i \pm \varepsilon_j$ generate $L_1$ (no. 4), so $Q(R) = L_1$ . Hence $Q(R^*) = L_1$ and consequently $P(R) = L_2$ (no. 4). By no. 4, $P(R)/Q(R)$ is isomorphic to $\mathbf{Z}/4\mathbf{Z}$ for $l$ odd, and to $\mathbf{Z}/2\mathbf{Z} \times \mathbf{Z}/2\mathbf{Z}$ for $l$ even. In the first case, $P(R)/Q(R)$ is generated by the canonical image of $\omega_l$ (and also by that of $\omega_{l-1}$ ). In the second case, $P(R)/Q(R)$ is generated by the canonical images of $\omega_{l-1}$ and $\omega_l$ . In both cases the connection index is 4.

(IX) and (X) In $\mathbf{R}^l$ , the orthogonal reflection $s_{\varepsilon_i, -\varepsilon_j}$ ( $i \neq j$ ) interchanges $\varepsilon_i$ and $\varepsilon_j$ and leaves invariant the $\varepsilon_k$ with $k$ distinct from $i$ and $j$ . The $s_{\varepsilon_i, -\varepsilon_j}$ generate a group $G_1$ isomorphic to the symmetric group $S_l$ . On the other hand, $s_{ij} = s_{\varepsilon_i, -\varepsilon_j} s_{\varepsilon_i, +\varepsilon_j}$ transforms $\varepsilon_i$ to $-\varepsilon_i$ , $\varepsilon_j$ to $-\varepsilon_j$ and leaves invariant the $\varepsilon_k$ with $k$ distinct from $i$ and $j$ . The $s_{ij}$ generate a group $G_2$ , the set of automorphisms $u$ of the vector space $\mathbf{R}^l$ such that $u(\varepsilon_i) = (-1)^{\nu_i} \varepsilon_i$ with $\prod_{i=1}^l (-1)^{\nu_i} = 1$ . The group $G_2$ is isomorphic to $(\mathbf{Z}/2\mathbf{Z})^{l-1}$ , and $G_2$ is normal in $W(R)$ , so $W(R)$ is isomorphic to the semi-direct product of $S_l$ by $(\mathbf{Z}/2\mathbf{Z})^{l-1}$ . Consequently, its order is $2^{l-1} l!$ .

The polynomial functions $t_i$ of no. 5 are invariant under $W(R)$ , and so is $t = \xi_1 \xi_2 \ldots \xi_l$ ; moreover, $t_l = t^2$ . Let $P(\xi_1, \ldots, \xi_l)$ be a polynomial invariant under $W(R)$ . Let $\xi_1^{\nu_1} \xi_2^{\nu_2} \ldots \xi_l^{\nu_l}$ be a monomial featuring in $P$ such that $\nu_i$ is odd; then $\nu_j$ is odd for all $j$ because the monomial $(-1)^{\nu_i + \nu_j} \xi_1^{\nu_1} \xi_2^{\nu_2} \ldots \xi_l^{\nu_l}$ features in $s_{ij}(P)$ , so $\nu_i + \nu_j \equiv 0 \pmod{2}$ and $\nu_j \equiv 1 \pmod{2}$ . Thus

P = P_1 + tP_2, where all the monomials featuring in P_1 and P_2 have only even exponents. Since P is invariant under the permutations of the $\xi_i$ , P_1 and P_2 have the same property, and so can be written as polynomials in $t_1, t_2, \ldots, t_l$ . This proves that the algebra $S(\mathbf{R}^l)^{W(R)}$ is generated by $t_1, t_2, \ldots, t_{l-1}, t$ . Moreover, the transcendence degree of the field of fractions of $S(\mathbf{R}^l)^{W(R)}$ is l, so $t_1, t_2, \ldots, t_{l-1}, t$ are algebraically independent. We conclude that the sequence of exponents, suitably ordered, is:

1, 3, 5, \ldots, 2l - 5, 2l - 3, l - 1.

Note that $l - 1$ appears twice if $l$ is even, and once if $l$ is odd.

(XI) The automorphisms of the Dynkin graph are those of the underlying graph. Thus: 1) If $l = 3$ , $A(R)/W(R)$ is isomorphic to $\mathbf{Z}/2\mathbf{Z}$ . 2) If $l = 4$ , every permutation of the terminal vertices defines an automorphism of the graph, so $A(R)/W(R)$ is isomorphic to $\mathfrak{S}_3$ . 3) If $l \geq 5$ , the chains starting at the ramification point have length 1, 1, and $l - 3 \geq 2$ . The only automorphism of the graph distinct from the identity thus corresponds to the automorphism $\varepsilon \in A(R)$ which interchanges $\alpha_{l-1}$ and $\alpha_l$ and leaves fixed the $\alpha_i$ for $1 \leq i \leq l - 2$ . Thus $A(R)/W(R)$ is isomorphic to $\mathbf{Z}/2\mathbf{Z}$ ; moreover, $A(R)$ is the semi-direct product of the group $G_1 \cong \mathfrak{S}_l$ defined in (IX) by the group $G_3$ consisting of the automorphisms $u$ of $\mathbf{R}^l$ such that $u(\varepsilon_i) = \pm \varepsilon_i$ for all $i$ .

If $l$ is even, $-1 \in W(R)$ , so $w_0 = -1$ . If $l$ is odd, $-1 \notin W(R)$ , so $A(R) = W(R) \times \{1, -1\}$ and $w_0 = -\varepsilon$ .

(XII) For $l$ even, $P(R^-)/Q(R^-)$ has three elements of order 2, namely $\omega_1, \omega_{l-1}$ and $\omega_l$ . Since $\omega_l$ (resp. $\omega_{l-1}$ ) interchanges the vertices corresponding to $\alpha_0$ and $\alpha_l$ (resp. $\alpha_{l-1}$ ), it interchanges those corresponding to $\alpha_1$ and $\alpha_{l-1}$ (resp. $\alpha_l$ ) and also those corresponding to $\alpha_j$ and $\alpha_{l-j}$ for

2 \leq j \leq l - 2.

We have $\omega_1 = \omega_l \omega_{l-1}$ .

For $l$ odd, $P(R^-)/Q(R^-)$ has two elements of order 4, namely $\omega_{l-1}$ and $\omega_l$ , and one element of order 2, equal to $\omega_1$ . Indeed, $\omega_1$ interchanges the vertices corresponding to $\alpha_0$ and $\alpha_1$ , so it leaves fixed the vertices corresponding to $\alpha_j$ for $2 \leq j \leq l - 2$ and is necessarily of order 2. Consequently $\omega_l$ is of order 4 and transforms the vertex corresponding to $\alpha_0$ (resp. $\alpha_l$ , resp. $\alpha_1$ , resp. $\alpha_{l-1}$ ) to that corresponding to $\alpha_l$ (resp. $\alpha_1$ , resp. $\alpha_{l-1}$ , resp. $\alpha_0$ ), and interchanges the vertices corresponding to $\alpha_j$ and $\alpha_{l-j}$ for $2 \leq j \leq l - 2$ . We have $\omega_1 = \omega_l^2$ and $\omega_{l-1} = \omega_l^3$ .

For $l \neq 4$ , the non-identity element of $A(R)/W(R)$ interchanges the vertices corresponding to $\alpha_{l-1}$ and $\alpha_l$ , and consequently interchanges the elements $\omega_{l-1}$ and $\omega_l$ of $P(R^-)/Q(R^-)$ . For $l$ odd, the automorphism of $P(R^-)/Q(R^-)$ thus obtained is the map $x \mapsto -x$ .

For $l = 4$ , $A(R)/W(R)$ can be identified with the group of permutations of $\{1, 3, 4\}$ and acts by permutations of the indices on $\{\omega_1, \omega_3, \omega_4\}$ .

9. System of type $F_4$

(I) Consider the group $L_2$ (no. 4) in $\mathbf{R}^4$ . Let $R$ be the set of $\alpha \in L_2$ such that $(\alpha|\alpha) = 1$ or $(\alpha|\alpha) = 2$ ; it contains the vectors

\pm \varepsilon_i,\quad \pm \varepsilon_i \pm \varepsilon_j\ (i < j),\quad \frac{1}{2}(\pm \varepsilon_1 \pm \varepsilon_2 \pm \varepsilon_3 \pm \varepsilon_4).

Conversely, if

\alpha \in R

, the coordinates of

\alpha

can only take the values

0, \pm \frac{1}{2}, \pm 1

(since

(\frac{3}{2})^2 > 2

); either these coordinates are all integers, giving the vectors

\pm \varepsilon_i, \pm \varepsilon_i \pm \varepsilon_j

, or they are all equal to

\pm \frac{1}{2}

, giving the vectors

\frac{1}{2}(\pm \varepsilon_1 \pm \varepsilon_2 \pm \varepsilon_3 \pm \varepsilon_4)

We show that, for $\alpha, \beta \in R$ , we have $2(\alpha|\beta)/(|\alpha|\alpha) \in \mathbf{Z}$ . If $\alpha = \pm \varepsilon_i$ or $\alpha = \frac{1}{2}(\pm \varepsilon_1 \pm \varepsilon_2 \pm \varepsilon_3 \pm \varepsilon_4)$ , then $(\alpha|\alpha) = 1$ and we have seen in no. 4 that $(\alpha|\beta) \in \frac{1}{2}\mathbf{Z}$ since $\alpha, \beta \in L_2$ . If $\alpha = \pm \varepsilon_i \pm \varepsilon_j$ , then $(\alpha|\alpha) = 2$ and we have seen in no. 4 that $(\alpha|\beta) \in \mathbf{Z}$ since $\alpha \in L_1$ and $\beta \in L_2$ . Hence, $R$ is a reduced root system in $\mathbf{R}^4$ (no. 4). The number of roots is $n = 8 + \binom{4}{2} 4 + 2^4 = 48$ .

(II) Give $\mathbf{R}^4$ the lexicographic order defined by the basis $(\varepsilon_1, \varepsilon_2, \varepsilon_3, \varepsilon_4)$ (§ 1, no. 7). In particular, we have $\varepsilon_1 > \varepsilon_2 > \varepsilon_3 > \varepsilon_4$ . The positive roots are

\varepsilon_i,\quad \varepsilon_i \pm \varepsilon_j\ (i < j),\quad \frac{1}{2}(\varepsilon_1 \pm \varepsilon_2 \pm \varepsilon_3 \pm \varepsilon_4).

The smallest root is

\alpha_3 = \varepsilon_4

. Among the roots belonging to

\mathbf{R}\varepsilon_3 + \mathbf{R}\varepsilon_4

but not to

\mathbf{R}\varepsilon_4

, the smallest is

\alpha_2 = \varepsilon_3 - \varepsilon_4

. Among the positive roots belonging to

\mathbf{R}\varepsilon_2 + \mathbf{R}\varepsilon_3 + \mathbf{R}\varepsilon_4

but not to

\mathbf{R}\varepsilon_3 + \mathbf{R}\varepsilon_4

, the smallest is

\alpha_1 = \varepsilon_2 - \varepsilon_3

. Among the positive roots not belonging to

\mathbf{R}\varepsilon_2 + \mathbf{R}\varepsilon_3 + \mathbf{R}\varepsilon_4

, the smallest is

\alpha_4 = \frac{1}{2}(\varepsilon_1 - \varepsilon_2 - \varepsilon_3 - \varepsilon_4)

. None of the

\alpha_i

is a sum of 2 positive roots. Hence,

(\alpha_1, \alpha_2, \alpha_3, \alpha_4)

is a basis of

R

(§ 1, no. 6, Cor. 1 to Prop. 19). We have

\| \alpha_1 \| = \| \alpha_2 \| = 2, \| \alpha_3 \| = \| \alpha_4 \| = 1, (\alpha_1|\alpha_2) = (\alpha_2|\alpha_3) = -1, (\alpha_3|\alpha_4) = -\frac{1}{2}, (\alpha_1|\alpha_3) = (\alpha_1|\alpha_4) = (\alpha_2|\alpha_4) = 0

. We see that the Dynkin graph of

R

is of type

F_4

, and hence is irreducible.

(III) We have $h = \frac{n}{l} = 12$ .

(IV) Let $\tilde{\alpha} = \varepsilon_1 + \varepsilon_2 = 2\alpha_1 + 3\alpha_2 + 4\alpha_3 + 2\alpha_4$ . The sum of the coordinates of $\tilde{\alpha}$ with respect to $(\alpha_i)$ is $11 = h - 1$ , so $\tilde{\alpha}$ is the highest root. We have $(\tilde{\alpha}|\alpha_1) = 1, (\tilde{\alpha}|\alpha_2) = (\tilde{\alpha}|\alpha_3) = (\tilde{\alpha}|\alpha_4) = 0$ .

The completed Dynkin graph is

!Dynkin graph of F_4

(V) The formula $\alpha^* = \frac{2\alpha}{(\alpha|\alpha)}$ gives for $R^-$ the set of vectors $\pm 2\varepsilon_i, \pm \varepsilon_i \pm \varepsilon_j, \pm \varepsilon_1 \pm \varepsilon_2 \pm \varepsilon_3 \pm \varepsilon_4$ . The Dynkin graph of $R^-$ is obtained from that of $R$ by the procedure explained in no. 2, and we see that $R^-$ is of type $F_4$ .

The roots not orthogonal to $\beta = \varepsilon_1$ are $\pm \varepsilon_1, \pm \varepsilon_1 \pm \varepsilon_j \ (j \geq 2)$ , and $\frac{1}{2}(\pm \varepsilon_1 \pm \varepsilon_2 \pm \varepsilon_3 \pm \varepsilon_4)$ : the number $n(\alpha|\beta) = 2(\alpha|\beta)$ is equal to $\pm 2$ for the first 14 of these roots and to $\pm 1$ for the last 16; thus, for $\Phi_R$ , the square of the length of $\beta$ is $4(14.4 + 16.1)^{-1} = \frac{1}{18}$ ; hence

\Phi_R(x, y) = \frac{(x|y)}{18}.

We now apply formula (18) of § 1, no. 12, with $x = y = \beta$ ; this gives

2 + 12.\frac{1}{4} + 16.\frac{1/4}{1} = \gamma(R).\frac{1}{18}

\gamma(R) = 2.3^4.

(VI) Calculating the fundamental weights gives

\begin{align*} \omega_1 &= \varepsilon_1 + \varepsilon_2 = 2\alpha_1 + 3\alpha_2 + 4\alpha_3 + 2\alpha_4 = \tilde{\alpha} \\ \omega_2 &= 2\varepsilon_1 + \varepsilon_2 + \varepsilon_3 = 3\alpha_1 + 6\alpha_2 + 8\alpha_3 + 4\alpha_4 \\ \omega_3 &= \frac{1}{2}(3\varepsilon_1 + \varepsilon_2 + \varepsilon_3 + \varepsilon_4) = 2\alpha_1 + 4\alpha_2 + 6\alpha_3 + 3\alpha_4 \\ \omega_4 &= \varepsilon_1 = \alpha_1 + 2\alpha_2 + 3\alpha_3 + 2\alpha_4. \end{align*}

(VII) The sum of the positive roots is

2\rho = 11\varepsilon_1 + 5\varepsilon_2 + 3\varepsilon_3 + \varepsilon_4 = 16\alpha_1 + 30\alpha_2 + 42\alpha_3 + 22\alpha_4.

(VIII) We have $Q(R) = L_2$ (no. 4), and $P(R) = Q(R)$ by (VI). Hence, the connection index is 1.

(IX) The family of exponents has 4 terms, and since $h = 12$ , it must contain the integers 1, 5, 7, 11, coprime to 12 (§ 1, no. 11, Prop. 30); consequently, these are all the exponents of $W(R)$ .

(X) and (XI) The only automorphism of the Dynkin graph is the identity, so $A(R) = W(R)$ and $w_0 = -1$ . Let $R'$ be the set of longest elements of $R$ , that is, the $\pm \varepsilon_i \pm \varepsilon_j$ ; $R'$ is the root system of type $D_4$ constructed in no. 8. Clearly, every element of $A(R)$ is an element of $A(R')$ . Conversely, an element of $A(R')$ leaves $L_1$ stable (since it is generated by $R'$ ), hence also its associated $L_2$ , and hence also $R$ . So $W(R) = A(R) = A(R')$ . By no. 8, $W(R)$ is the semi-direct product of $S_3$ by $W(R')$ , $W(R')$ itself being the semi-direct product of $S_4$ by $(\mathbf{Z}/2\mathbf{Z})^3$ . The order of $W(R)$ is $3!4!2^3 = 2^7.3^2$ .

10. System of type e_8

(I) Consider the group L_3 (no. 4) in $\mathbf{R}^8$ . Let R be the set of $\alpha \in L_3$ such that $(\alpha|\alpha) = 2$ ; it contains the vectors

\pm \varepsilon_i \pm \varepsilon_j \quad (i < j), \quad \frac{1}{2} \sum_{i=1}^8 (-1)^{\nu(i)} \varepsilon_i \quad (\sum_{i=1}^8 \nu(i) \text{ even}).

Conversely, if an element $\alpha \in L_3$ is such that $(\alpha|\alpha) = 2$ , its coordinates must be among the values $0, \pm \frac{1}{2}, \pm 1$ : by no. 4, either these coordinates are all integers, giving the vectors $\pm \varepsilon_i \pm \varepsilon_j$ , or they are all equal to $\pm \frac{1}{2}$ with an integer sum, giving the vectors

\frac{1}{2} \sum_{i=1}^8 (-1)^{\nu(i)} \varepsilon_i

with $\sum_{i=1}^8 \nu(i)$ even.

We have seen (no. 4) that $(\alpha|\beta) \in \mathbf{Z}$ for all $\alpha, \beta \in L_3$ . Hence, R is a reduced root system. The number of roots is $n = \binom{8}{2} . 4 + 2^7 = 240$ .

(II) Let $\rho$ be the vector $(0, 1, 2, 3, 4, 5, 6, 23)$ of $L_3$ . No element of R is orthogonal to $\rho$ (this is clear for the $\pm \varepsilon_i \pm \varepsilon_j$ ; if $\frac{1}{2} \sum_{i=1}^8 (-1)^{\nu(i)} \varepsilon_i$ were orthogonal to $\rho$ , we would have $\sum_{i=1}^6 i(-1)^{\nu(i+1)} + 23(-1)^{\nu(8)} = 0$ , which is impossible since $\sum_{i=1}^6 i < 23$ ). Hence (§ 1, no. 7, Cor. 2 of Prop. 20) the $\alpha \in R$ such that $(\alpha|\rho) > 0$ are the positive roots relative to a certain chamber. These roots are the $\pm \varepsilon_i + \varepsilon_j \ (i < j)$ , and the

\frac{1}{2} (\varepsilon_8 + \sum_{i=1}^7 (-1)^{\nu(i)} \varepsilon_i)

with $\sum_{i=1}^7 \nu(i)$ even. We have $(\alpha|\rho) \in \mathbf{Z}$ for all $\alpha \in R$ (no. 4), and $(\alpha|\rho)$ is equal to 1 for the following roots:

\begin{align*} \alpha_1 &= \frac{1}{2} (\varepsilon_1 + \varepsilon_8) - \frac{1}{2} (\varepsilon_2 + \varepsilon_3 + \varepsilon_4 + \varepsilon_5 + \varepsilon_6 + \varepsilon_7), \\ \alpha_2 &= \varepsilon_1 + \varepsilon_2, \ \alpha_3 = \varepsilon_2 - \varepsilon_1, \ \alpha_4 = \varepsilon_3 - \varepsilon_2, \ \alpha_5 = \varepsilon_4 - \varepsilon_3, \\ \alpha_6 &= \varepsilon_5 - \varepsilon_4, \ \alpha_7 = \varepsilon_6 - \varepsilon_5, \ \alpha_8 = \varepsilon_7 - \varepsilon_6. \end{align*}

and these eight vectors form a basis of $\mathbf{R}^8$ . By § 1, no. 6, Cor. 1 of Prop. 19, $(\alpha_1, \alpha_2, \ldots, \alpha_8)$ is a basis of R for which the positive roots are those which have been defined above. We have

(\alpha_4|\alpha_5) = (\alpha_5|\alpha_6) = (\alpha_6|\alpha_7) = (\alpha_7|\alpha_8) = (\alpha_4|\alpha_2) = (\alpha_4|\alpha_3) = (\alpha_3|\alpha_1) = -1,

and $(\alpha_i|\alpha_j) = 0$ for all other pairs of indices. Hence, the Dynkin graph of R is of type E_8, and R is irreducible.

(III) We have $h = \frac{n}{8} = 30$ .

(IV) Let

\tilde{\alpha} = \varepsilon_7 + \varepsilon_8 = 2\alpha_1 + 3\alpha_2 + 4\alpha_3 + 6\alpha_4 + 5\alpha_5 + 4\alpha_6 + 3\alpha_7 + 2\alpha_8,

which is a root. The sum of its coordinates with respect to

(\alpha_i)

29 = h - 1

, so

\tilde{\alpha}

is the highest root. It is orthogonal to all the

\alpha_i

except

\alpha_8

, and

(\tilde{\alpha}|\alpha_8) = 1

. Hence, the completed Dynkin graph is:

!Dynkin graph of E_8

(V) Since $(\alpha|\alpha) = 2$ for all $\alpha \in R$ , we have $R^- = R$ .

For $\Phi_R$ , the squared length of the roots is $\frac{1}{30}$ (§ 1, no. 12). Hence, $\Phi_R(x, y) = (x|y)/60$ and $\gamma(R) = 900$ (§ 1. no. 12, formula (20)).

(VI) Calculating the fundamental weights gives

\begin{align*} \omega_1 &= 2\varepsilon_8 = 4\alpha_1 + 5\alpha_2 + 7\alpha_3 + 10\alpha_4 + 8\alpha_5 + 6\alpha_6 + 4\alpha_7 + 2\alpha_8 \\ \omega_2 &= \frac{1}{2}(\varepsilon_1 + \varepsilon_2 + \varepsilon_3 + \varepsilon_4 + \varepsilon_5 + \varepsilon_6 + \varepsilon_7 + 5\varepsilon_8) \\ &= 5\alpha_1 + 8\alpha_2 + 10\alpha_3 + 15\alpha_4 + 12\alpha_5 + 9\alpha_6 + 6\alpha_7 + 3\alpha_8 \\ \omega_3 &= \frac{1}{2}(-\varepsilon_1 + \varepsilon_2 + \varepsilon_3 + \varepsilon_4 + \varepsilon_5 + \varepsilon_6 + \varepsilon_7 + 7\varepsilon_8) \\ &= 7\alpha_1 + 10\alpha_2 + 14\alpha_3 + 20\alpha_4 + 16\alpha_5 + 12\alpha_6 + 8\alpha_7 + 4\alpha_8 \\ \omega_4 &= \varepsilon_3 + \varepsilon_4 + \varepsilon_5 + \varepsilon_6 + \varepsilon_7 + 5\varepsilon_8 \\ &= 10\alpha_1 + 15\alpha_2 + 20\alpha_3 + 30\alpha_4 + 24\alpha_5 + 18\alpha_6 + 12\alpha_7 + 6\alpha_8 \\ \omega_5 &= \varepsilon_4 + \varepsilon_5 + \varepsilon_6 + \varepsilon_7 + 4\varepsilon_8 \\ &= 8\alpha_1 + 12\alpha_2 + 16\alpha_3 + 24\alpha_4 + 20\alpha_5 + 15\alpha_6 + 10\alpha_7 + 5\alpha_8 \\ \omega_6 &= \varepsilon_5 + \varepsilon_6 + \varepsilon_7 + 3\varepsilon_8 \\ &= 6\alpha_1 + 9\alpha_2 + 12\alpha_3 + 18\alpha_4 + 15\alpha_5 + 12\alpha_6 + 8\alpha_7 + 4\alpha_8 \\ \omega_7 &= \varepsilon_6 + \varepsilon_7 + 2\varepsilon_8 \\ &= 4\alpha_1 + 6\alpha_2 + 8\alpha_3 + 2\alpha_4 + 10\alpha_5 + 8\alpha_6 + 6\alpha_7 + 3\alpha_8 \\ \omega_8 &= \varepsilon_7 + \varepsilon_8 \\ &= 5\alpha_1 + 8\alpha_2 + 10\alpha_3 + 15\alpha_4 + 12\alpha_5 + 9\alpha_6 + 6\alpha_7 + 3\alpha_8 = \tilde{\alpha}. \end{align*}

(VII) Half the sum of the positive roots is the sum of the fundamental weights (§ 1, no. 10, Prop. 29); this gives

\begin{align*} \rho &= \varepsilon_2 + 2\varepsilon_3 + 3\varepsilon_4 + 4\varepsilon_5 + 5\varepsilon_6 + 6\varepsilon_7 + 23\varepsilon_8 \\ &= 46\alpha_1 + 68\alpha_2 + 91\alpha_3 + 135\alpha_4 + 110\alpha_5 + 84\alpha_6 + 57\alpha_7 + 29\alpha_8. \end{align*}

(VIII) The group Q(R) is generated by the $\varepsilon_i \pm \varepsilon_j$ and $\frac{1}{2} \sum_{i=1}^8 \varepsilon_i$ , and is equal to L_3 (no. 4). Hence P(R), which is associated to Q(R̄) = Q(R) = L_3, is L_3 (no. 4). The connection index is 1.

(IX) The family of exponents has 8 terms, and since $h = 30$ , the integers 1, 7, 11, 13, 17, 19, 23, 29, coprime to 30, must feature in this family; consequently, these are the exponents of W(R).

(X) From (IX) and Chap. V. § 6, no. 2, Cor. 1 of Prop. 3, it follows that the order of W(R) is

2.8.12.14.18.20.24.30 = 2^{14}.3^5.5^2.7.

(XI) The only automorphism of the Dynkin graph is the identity since the three chains issuing from the ramification point have distinct lengths. Hence, $A(R) = W(R)$ and $w_0 = -1$ .

11. System of type e_7

(I) and (II) Let $E = \mathbf{R}^8$ , and let R_8 be the root system in E constructed in no. 10. Let V be the hyperplane in E generated by the roots $\alpha_1, \ldots, \alpha_7$ of R_8; it is orthogonal to the eighth fundamental weight $\omega = \varepsilon_7 + \varepsilon_8$ of R_8.

Let $R = R_8 \cap V$ . Then R is a reduced root system with basis $(\alpha_1, \ldots, \alpha_7)$ , cf. § 1, no. 7, Cor. 4 of Prop. 20; hence, this system is of type E_7. Its elements are:

\pm \varepsilon_i \pm \varepsilon_j \quad (1 \leq i \leq j \leq 6), \quad \pm (\varepsilon_7 - \varepsilon_8),

\pm \frac{1}{2} (\varepsilon_7 - \varepsilon_8 + \sum_{i=1}^8 (-1)^{\nu(i)} \varepsilon_i) \quad \text{with } \sum_{i=1}^8 \nu(i) \text{ odd}.

The number of roots is $n = 2 + \binom{6}{2}.4 + 2^6 = 126$ . The positive roots are

\pm \varepsilon_i + \varepsilon_j \quad (1 \leq i < j \leq 6), \quad -\varepsilon_7 + \varepsilon_8,

\frac{1}{2} (-\varepsilon_7 + \varepsilon_8 + \sum_{i=1}^6 (-1)^{\nu(i)} \varepsilon_i) \quad \text{with } \sum_{i=1}^6 \nu(i) \text{ odd}.

(III) We have $h = \frac{n}{l} = 18$ .

(IV) Let $\tilde{\alpha} = \varepsilon_8 - \varepsilon_7 = 2\alpha_1 + 2\alpha_2 + 3\alpha_3 + 4\alpha_4 + 3\alpha_5 + 2\alpha_6 + \alpha_7$ , which is a root. The sum of its coordinates with respect to $(\alpha_i)$ is 17 = $h - 1$ . It is therefore the highest root. It is orthogonal to $\alpha_i$ for $2 \leq i \leq 7$ , and $(\tilde{\alpha}|\alpha_1) = 1$ . The completed Dynkin graph is

!Dynkin graph for E_7

(V) Since $(\alpha|\alpha) = 2$ for all $\alpha \in R$ , we have $R^- = R$ . For $\Phi_R$ , the squared length of the roots is $\frac{1}{18}$ , so

\Phi_R(x, y) = (x|y)/36, \quad \text{and} \quad \gamma(R) = 2^2 \cdot 3^4

(§ 1. no. 12, formula (20)).

(VI) Calculating the fundamental weights gives

\begin{align*} \omega_1 &= \varepsilon_8 - \varepsilon_7 = 2\alpha_1 + 2\alpha_2 + 3\alpha_3 + 4\alpha_4 + 3\alpha_5 + 2\alpha_6 + \alpha_7 \\ \omega_2 &= \frac{1}{2}(\varepsilon_1 + \varepsilon_2 + \varepsilon_3 + \varepsilon_4 + \varepsilon_5 + \varepsilon_6 - 2\varepsilon_7 + 2\varepsilon_8) \\ &= \frac{1}{2}(4\alpha_1 + 7\alpha_2 + 8\alpha_3 + 12\alpha_4 + 9\alpha_5 + 8\alpha_6 + 3\alpha_7) \\ \omega_3 &= \frac{1}{2}(-\varepsilon_1 + \varepsilon_2 + \varepsilon_3 + \varepsilon_4 + \varepsilon_5 + \varepsilon_6 - 3\varepsilon_7 + 3\varepsilon_8) \\ &= 3\alpha_1 + 4\alpha_2 + 6\alpha_3 + 8\alpha_4 + 6\alpha_5 + 4\alpha_6 + 2\alpha_7 \\ \omega_4 &= \varepsilon_3 + \varepsilon_4 + \varepsilon_5 + \varepsilon_6 + 2(\varepsilon_8 - \varepsilon_7) \\ &= 4\alpha_1 + 6\alpha_2 + 8\alpha_3 + 12\alpha_4 + 9\alpha_5 + 6\alpha_6 + 3\alpha_7 \\ \omega_5 &= \varepsilon_4 + \varepsilon_5 + \varepsilon_6 + \frac{3}{2}(\varepsilon_8 - \varepsilon_7) \\ &= \frac{1}{2}(6\alpha_1 + 9\alpha_2 + 12\alpha_3 + 18\alpha_4 + 15\alpha_5 + 10\alpha_6 + 5\alpha_7) \\ \omega_6 &= \varepsilon_5 + \varepsilon_6 - \varepsilon_7 + \varepsilon_8 \\ &= 2\alpha_1 + 3\alpha_2 + 4\alpha_3 + 6\alpha_4 + 5\alpha_5 + 4\alpha_6 + 2\alpha_7 \\ \omega_7 &= \varepsilon_6 + \frac{1}{2}(\varepsilon_8 - \varepsilon_7) \\ &= \frac{1}{2}(2\alpha_1 + 3\alpha_2 + 4\alpha_3 + 6\alpha_4 + 5\alpha_5 + 4\alpha_6 + 3\alpha_7). \end{align*}

(VII) The sum $2\rho$ of the positive roots is $2 \sum_{i=1}^7 \omega_i$ (§ 1. no. 10. Prop. 29), so

\begin{align*} 2\rho &= 2\varepsilon_2 + 4\varepsilon_3 + 6\varepsilon_4 + 8\varepsilon_5 + 10\varepsilon_6 - 17\varepsilon_7 + 17\varepsilon_8 \\ &= 34\alpha_1 + 49\alpha_2 + 66\alpha_3 + 96\alpha_4 + 75\alpha_5 + 52\alpha_6 + 27\alpha_7. \end{align*}

(VIII) By no. 10 (VIII) and § 1, no. 10. Prop. 28, we have

Q(R) = Q(R_8) \cap V = L_3 \cap V \quad \text{and} \quad P(R) = p(P(R_8)) = p(L_3),

where

p

denotes the orthogonal projection of

E

onto

V

. The group

Q(R)

has basis

(\alpha_1, \ldots, \alpha_7)

; the group

P(R)

is generated by

Q(R)

and

p(\alpha_8) = \alpha_8 - \frac{1}{2}\omega.

We have $\omega \in P(R_8)$ , $\frac{1}{2}\omega \notin P(R_8)$ . so $2p(\alpha_8) \in Q(R)$ and $p(\alpha_8) \notin Q(R)$ . Thus, $P(R)/Q(R)$ is isomorphic to $\mathbf{Z}/2\mathbf{Z}$ and is generated by, for example, the image of $\omega_7$ .

The connection index is 2.

(IX) The sequence of exponents of $W(R)$ has 7 terms. The numbers 1, 5, 7, 11, 13, 17, coprime to $h = 18$ , feature in this sequence. The last exponent $m$ must be such that $m + m = 18$ (Chap. V, § 6, no. 2, formula (2)). Hence, the sequence of exponents is

1, 5, 7, 9, 11, 13, 17.

(X) From (IX) and Chap. V, § 6, no. 2, Cor. 1 of Prop. 3, it follows that the order of $W(R)$ is

2.6.8.10.12.14.18 = 2^{10}.3^4.5.7.

(XI) The only automorphism of the Dynkin graph is the identity, so $A(R) = W(R)$ and $w_0 = -1$ .

(XII) $P(R^-)/Q(R^-)$ has only one non-identity element. It interchanges the vertices corresponding to $\alpha_0$ and $\alpha_7$ , $\alpha_1$ and $\alpha_6$ , $\alpha_3$ and $\alpha_5$ , and leaves $\alpha_2$ and $\alpha_4$ fixed.

12. System of type e_6

(I) and (II) Let $E = \mathbf{R}^8$ , and let $R_8$ be the root system in $E$ constructed in no. 10. Let $V$ be the vector subspace of $E$ generated by the roots $\alpha_1, \ldots, \alpha_6$ of $R_8$ ; this is the orthogonal complement of the plane generated by the last two fundamental weights $\omega = \varepsilon_7 + \varepsilon_8$ and $\pi = \varepsilon_6 + \varepsilon_7 + 2\varepsilon_8$ of $R_8$ .

Let $R = R_8 \cap V$ . This is a reduced root system with basis $(\alpha_1, \ldots, \alpha_6)$ , and hence of type $E_6$ . Its elements are:

\pm \varepsilon_i \pm \varepsilon_j \quad (1 \leq i < j \leq 5),

\pm \frac{1}{2} (\varepsilon_8 - \varepsilon_7 - \varepsilon_6 + \sum_{i=1}^5 (-1)^{\nu(i)} \varepsilon_i) \text{ with } \sum_{i=1}^5 \nu(i) \text{ even}.

The number of roots is $n = \binom{5}{2}.4 + 2^5 = 72$ . The positive roots are

\pm \varepsilon_i + \varepsilon_j \quad (1 \leq i < j \leq 5),

\frac{1}{2} (\varepsilon_8 - \varepsilon_7 - \varepsilon_6 + \sum_{i=1}^5 (-1)^{\nu(i)} \varepsilon_i) \text{ with } \sum_{i=1}^5 \nu(i) \text{ even}.

(III) We have $h = \frac{n}{6} = 12$ .

(IV) Let

\tilde{\alpha} = \frac{1}{2} (\varepsilon_1 + \varepsilon_2 + \varepsilon_3 + \varepsilon_4 + \varepsilon_5 - \varepsilon_6 - \varepsilon_7 + \varepsilon_8)

= \alpha_1 + 2\alpha_2 + 2\alpha_3 + 3\alpha_4 + 2\alpha_5 + \alpha_6,

which is a root. The sum of its coordinates with respect to $(\alpha_i)$ is $11 = h - 1$ , so $\tilde{\alpha}$ is the highest root. It is orthogonal to $\alpha_1, \alpha_3, \alpha_4, \alpha_5, \alpha_6$ , and $(\tilde{\alpha}|\alpha_2) = 1$ . The completed Dynkin graph is

!Dynkin graph

(V) Since $(\alpha|\alpha) = 2$ for all $\alpha \in R$ , we have $R' = R$ . For $\Phi_R$ , the squared length of the roots is $\frac{1}{12}$ , so

\Phi_R(x, y) = (x|y)/24, \quad \text{and} \quad \gamma(R) = 144.

(VI) Calculating the fundamental weights gives:

\begin{align*} \omega_1 &= \frac{2}{3}(\varepsilon_8 - \varepsilon_7 - \varepsilon_6) = \frac{1}{3}(4\alpha_1 + 3\alpha_2 + 5\alpha_3 + 6\alpha_4 + 4\alpha_5 + 2\alpha_6) \\ \omega_2 &= \frac{1}{2}(\varepsilon_1 + \varepsilon_2 + \varepsilon_3 + \varepsilon_4 + \varepsilon_5 - \varepsilon_6 - \varepsilon_7 + \varepsilon_8) \\ &= \alpha_1 + 2\alpha_2 + 2\alpha_3 + 3\alpha_4 + 2\alpha_5 + \alpha_6 = \tilde{\alpha} \\ \omega_3 &= \frac{5}{6}(\varepsilon_8 - \varepsilon_7 - \varepsilon_6) + \frac{1}{2}(-\varepsilon_1 + \varepsilon_2 + \varepsilon_3 + \varepsilon_4 + \varepsilon_5) \\ &= \frac{1}{3}(5\alpha_1 + 6\alpha_2 + 10\alpha_3 + 12\alpha_4 + 8\alpha_5 + 4\alpha_6) \\ \omega_4 &= \varepsilon_3 + \varepsilon_4 + \varepsilon_5 - \varepsilon_6 - \varepsilon_7 + \varepsilon_8 \\ &= 2\alpha_1 + 3\alpha_2 + 4\alpha_3 + 6\alpha_4 + 4\alpha_5 + 2\alpha_6 \\ \omega_5 &= \frac{2}{3}(\varepsilon_8 - \varepsilon_7 - \varepsilon_6) + \varepsilon_4 + \varepsilon_5 \\ &= \frac{1}{3}(4\alpha_1 + 6\alpha_2 + 8\alpha_3 + 12\alpha_4 + 10\alpha_5 + 5\alpha_6) \\ \omega_6 &= \frac{1}{3}(\varepsilon_8 - \varepsilon_7 - \varepsilon_6) + \varepsilon_5 \\ &= \frac{1}{3}(2\alpha_1 + 3\alpha_2 + 4\alpha_3 + 6\alpha_4 + 5\alpha_5 + 4\alpha_6). \end{align*}

(VII) Half the sum of the positive roots is $\sum_{i=1}^6 \omega_i$ , so

\begin{align*} \rho &= \varepsilon_2 + 2\varepsilon_3 + 3\varepsilon_4 + 4\varepsilon_5 + 4(\varepsilon_8 - \varepsilon_7 - \varepsilon_6) \\ &= 8\alpha_1 + 11\alpha_2 + 15\alpha_3 + 21\alpha_4 + 15\alpha_5 + 8\alpha_6. \end{align*}

(VIII) By no. 10 (VIII) and § 4, no. 10, Prop. 28,

Q(R) = Q(R_8) \cap V = L_3 \cap V \quad \text{and} \quad P(R) = p(P(R_8)) = p(L_3),

where $p$ denotes the orthogonal projection of $E$ onto $V$ . We have

p(\alpha_7) = \alpha_7 - \frac{2}{3}\pi + \omega, \quad p(\alpha_8) = \alpha_8 + \pi - 2\omega.

The group

Q(R)

has basis

(\alpha_1, \ldots, \alpha_6)

. The group

P(R)

is generated by

Q(R)

and

p(\alpha_7)

, since

p(\alpha_8) \in P(R_8) \cap V = Q(R_8) \cap V = Q(R)

. We have

3p(\alpha_7) \in Q(R)

and

p(\alpha_7) \notin Q(R)

. Hence, the group

P(R)/Q(R)

is isomorphic to

\mathbf{Z}/3\mathbf{Z}

; it is generated, for example, by the image of

\omega_6

The connection index is 3.

(IX) and (X) By § 2, no. 4, Prop. 7, the order of the Weyl group is $6!1.2.2.3.2.1.3 = 2^7.3^4.5$ . The sequence of exponents has 6 terms between 1 and 11, and contains the integers 1, 5, 7, 11 which are coprime to 12. The other exponents $m, m'$ are integers such that

m + m' = 12,

(m + 1)(m' + 1)(1 + 1)(5 + 1)(7 + 1)(11 + 1) = 2^7.3^4.5,

in view of Chap. V, § 6, no. 2, formula (2) and Cor. 1 of Prop. 3. The second relation gives

(m + 1)(m' + 1) = 45

, and since

m + m' + 2 = 14

, we obtain

m = 4, m' = 8

. Hence, the sequence of exponents is

1, 4, 5, 7, 8, 11.

(XI) and (XII) Since the roots all have the same length, the automorphisms of the Dynkin graph are those of the underlying graph. Apart from the identity, there is only the automorphism $\varepsilon$ which transforms $\alpha_1, \alpha_3, \alpha_4, \alpha_5, \alpha_6, \alpha_2$ into $\alpha_6, \alpha_5, \alpha_4, \alpha_3, \alpha_1, \alpha_2$ , respectively. Hence, $A(R)/W(R)$ is isomorphic to $\mathbf{Z}/2\mathbf{Z}$ ; since $-1 \in W(R)$ (Chap. V, § 6, no. 2, Cor. 3 of Prop. 3), $A(R)$ is isomorphic to $W(R) \times \{1, -1\}$ and $w_0$ can be identified with $-\varepsilon$ . It follows that the non-identity element of $A(R)/W(R)$ defines the automorphism $x \mapsto -x$ of $P(R^-)/Q(R^-)$ .

Moreover, $P(R^-)/Q(R^-)$ has two non-identity elements of order 3. They define the two automorphisms of order 3 of the completed Dynkin graph.

13. System of type g_2

(I) Let $E$ be the hyperplane in $\mathbf{R}^3$ with equation

\xi_1 + \xi_2 + \xi_3 = 0.

Let

R

be the set of

\alpha \in L_0 \cap V

such that

(\alpha|\alpha) = 2

(\alpha|\alpha) = 6

. The elements of

R

are

\pm (\varepsilon_1 - \varepsilon_2), \quad \pm (\varepsilon_1 - \varepsilon_3), \quad \pm (\varepsilon_2 - \varepsilon_3), \quad \pm (2\varepsilon_1 - \varepsilon_2 - \varepsilon_3),

\pm (2\varepsilon_2 - \varepsilon_1 - \varepsilon_3), \quad \pm (2\varepsilon_3 - \varepsilon_1 - \varepsilon_2).

Then, R generates V, and $\frac{2(\alpha|\beta)}{(\beta|\beta)} \in \mathbf{Z}$ for all $\alpha, \beta \in R$ : this is clear if $\beta = \pm (\varepsilon_i - \varepsilon_j)$ with $i \neq j$ ; if $\beta = 2\varepsilon_1 - \varepsilon_2 - \varepsilon_3$ for example, we have $(\alpha|\beta) \in 3\mathbf{Z}$ , and again our assertion holds. Hence, R is a reduced root system in V. The number of roots is $n = 12$ .

(II) Put $\alpha_1 = \varepsilon_1 - \varepsilon_2, \alpha_2 = -2\varepsilon_1 + \varepsilon_2 + \varepsilon_3$ . Then the roots are

\pm \alpha_1, \quad \pm (\alpha_1 + \alpha_2), \quad \pm (2\alpha_1 + \alpha_2), \quad \pm \alpha_2,

\pm (3\alpha_1 + \alpha_2), \quad \pm (3\alpha_1 + 2\alpha_2).

Hence, $(\alpha_1, \alpha_2)$ is a basis of R. We have $\| \alpha_1 \| ^2 = 2, \| \alpha_2 \| ^2 = 6, (\alpha_1|\alpha_2) = -3$ , so R is a system of type G_2. The positive roots are $\alpha_1, \alpha_1 + \alpha_2, 2\alpha_1 + \alpha_2, 3\alpha_1 + \alpha_2, 3\alpha_1 + 2\alpha_2$ .

(III) We have $h = \frac{n}{2} = 6$ .

(IV) The highest root is $\tilde{\alpha} = 3\alpha_1 + 2\alpha_2 = -\varepsilon_1 - \varepsilon_2 + 2\varepsilon_3$ . We have $(\tilde{\alpha}|\alpha_1) = 0, (\tilde{\alpha}|\alpha_2) = 3$ . The completed Dynkin graph is

!Dynkin graph

(V) The inverse system is the following set of vectors:

\pm \alpha_1, \quad \pm (\alpha_1 + \alpha_2), \quad \pm (2\alpha_1 + \alpha_2), \quad \pm \frac{1}{3}\alpha_2,

\pm \frac{1}{3}(3\alpha_1 + \alpha_2), \quad \pm \frac{1}{3}(3\alpha_1 + 2\alpha_2).

There are 10 roots not orthogonal to $\alpha_1$ ; we have $n(\beta, \alpha_1) = \pm 1$ for 4 of these roots, $n(\beta, \alpha_1) = \pm 3$ for 4 others, and $n(\beta, \alpha_1) = \pm 2$ for $\beta = \pm \alpha_1$ . Hence, the squared length of $\alpha_1$ for $\Phi_R$ is $4(4.1 + 4.9 + 2.4)^{-1} = \frac{1}{12}$ . Hence, $\Phi_R(x, y) = (x|y)/24$ . We now apply formula (18) of § 1, no. 12, with $x = y = \alpha_1$ ; this gives

2 + 4.\frac{1}{4} + 4.\frac{1}{4} = \gamma(R).\frac{1}{12}

so $\gamma(R) = 48$ .

(VI) and (VII) Half the sum of the positive roots is

\rho = 5\alpha_1 + 3\alpha_2.

The fundamental weights $\omega_1$ and $\omega_2$ are orthogonal to $\alpha_2$ and $\alpha_1$ , hence proportional to $2\alpha_1 + \alpha_2$ and $3\alpha_1 + 2\alpha_2$ . We have

\omega_1 + \omega_2 = \rho = 5\alpha_1 + 3\alpha_2 = (2\alpha_1 + \alpha_2) + (3\alpha_1 + 2\alpha_2).

Hence,

\omega_1 = 2\alpha_1 + \alpha_2, \quad \omega_2 = 3\alpha_1 + 2\alpha_2 = \tilde{\alpha}.

(VIII) Q(R) is generated by $\varepsilon_1 - \varepsilon_2$ and $\varepsilon_1 - \varepsilon_3$ , for example. By (VI) and (VII), P(R) = Q(R). The connection index is 1.

(IX) The family of exponents has 2 terms; since 1 and $h - 1 = 5$ are exponents, they are the only ones.

(X) We have $(\overline{\alpha_1, \alpha_2}) = \frac{5\pi}{6}$ , so W(R) is isomorphic to the dihedral group of order 12.

(XI) The only automorphism of the Dynkin diagram is the identity, so $\Lambda(R) = W(R)$ and $w_0 = -1$ .

14. Irreducible non-reduced root systems

The irreducible, non-reduced root systems can be obtained from the irreducible, reduced systems by using Props. 13 and 14 of § 1. no. 4. For each integer $l \geq 1$ , there exists, up to isomorphism, a unique irreducible, non-reduced root system of rank $l$ : let R be a root system of type B_l, A the set of shortest roots of R; take the union of R and 2A. With the notation of no. 5, we obtain the $2l(l+1)$ vectors

\pm \varepsilon_i,\ \pm 2\varepsilon_i,\ \varepsilon_i \pm \varepsilon_j\ (i < j).

Exercises

If R is a root system, denote by $W^+(R)$ the set of elements of $W(R)$ of determinant 1.

See the exercises for § 4.