Introduction

This chapter describes Magma functions for computing with with finite real root data. It was implemented by Scott Murray and Don Taylor. Parts of it are modelled on the Chevie package in GAP [GHLetalchar+96].

Root data are essential in the theories of finite Coxeter groups (Chapter COXETER GROUPS), and groups of Lie type (Chapter GROUPS OF LIE TYPE). A root datum essentially describes a collection of reflections which generate a group.

Subsections

• Definition of a root datum
• Positive and simple roots
• Classification of root data

Definition of a root datum

Our description of root data follows [Dem65] and [Car93] with two modifications: we have expanded the definition to include noncrystallographic root data and our reflections act on the right as in customary in Magma.

Let X and Y be real vector spaces with a bilinear pairing < o , o >:X x Y -> R that identifies Y with the dual of X. We call X the root space and Y the coroot space. Assume we have a basis {e_1, ..., e_n} for X and a dual basis {omega_1, ..., omega_n} for Y, so that < e_i, omega_j >=delta_(ij). Let Phi be a finite subset of X and suppose that for each alpha in Phi we have a corresponding alpha^star in Y; set Phi^star={alpha^star | alpha in Phi}. We call the elements of Phi roots and the elements of Phi^star coroots.

Given a root alpha we define the linear map s_alpha:X -> X by

x s_alpha= x - < x, alpha^star >alpha

and similarly the linear map s_alpha^star:Y -> Y by

y s_alpha^star= y - < alpha, y >alpha^star.

These maps are called reflections if one of the following equivalent properties hold: < alpha, alpha^star >=2; (s_alpha)^2=1; < xs_alpha, ys_alpha^star > = < x, y > for all x in X and y in Y; alpha s_alpha= - alpha.

We say that X, Phi, Y, Phi^star form a root datum if the following are satisfied for every root alpha in Phi:

• s_alpha and s_alpha^star are reflections;
• Phi is closed under the action of s_alpha; and
• Phi^star is closed under the action of s_alpha^star.

For many applications to Lie theory, we need to modify this definition slightly: we replace the R-spaces X and Y with standard integer lattices, and we require that the image of the pairing lies in Z, ie. < o , o >:X x Y -> Z. Since we can form R-spaces X_R=X otimes_Z R and Y_R=Y otimes_Z R, this is equivalent to the previous definition, with the added conditions that every root must be a Z-linear combination of e_1, ..., e_n, and every coroot must be a Z-linear combination of omega_1, ..., omega_n. We call such root data crystallographic.

Positive and simple roots

A subset Delta of Phi is called a set of simple roots if

• Delta is a basis for the span of the roots R Phi <= X_R.
• Phi = Phi^ + union Phi^ -, where Phi^ + is the set of linear combinations of elements of Delta with nonnegative coefficients, and Phi^ - = - Phi^ +.

Every root datum has a set of simple roots. Moreover, if Delta_1 and Delta_2 are two sets of simple roots for Phi, then Delta_2 = Delta_1 w for a unique element w in the group generated by the reflections s_alpha, alpha in Phi.

The rank of the root system is the size of Delta, ie. the dimension of the subspace R Phi. In a crystallographic root system, every root is a Z-linear combination of the simple roots and the height of a root is the sum of its coordinates with respect to the simple roots.

We can define a reflection invariant bilinear form on X_R by

(x, y) = sum_(alpha in Phi) < x, alpha^star > < y, alpha^star >.

Let alpha^v=2alpha/(alpha, alpha). Given alpha, beta in Phi, we have

< alpha, beta^star > = (alpha, beta^v) = frac(2(alpha, beta))((beta, beta)).

The Cartan matrix of the indexed set {@ alpha_1, alpha_2, ..., alpha_n @} of simple roots is

C = [< alpha_i, alpha_j^star >].

Classification of root data

In this section we describe the classification of semisimple finite real root data. If the span of the roots, R Phi, is all of X_R, then we say the root datum is semisimple. Every root datum contains a radical, which is a trivial root datum (ie. a root datum with no roots, even though the root space may be nontrivial). The quotient of a root datum by its radical is semisimple.

Given a semisimple root datum we define a graph with vertices corresponding to the simple roots and an edge connecting the vertices for alpha and beta whenever < alpha, beta^star > is nonzero. A semisimple root datum is a direct sum of the root subdata corresponding to the connected components of this graph.

We obtain the Dynkin diagram by adding further information to this graph:

• In the crystallographic case, the vertices for alpha and beta are linked by < alpha, beta^star > < beta, alpha^star > edges; if the lengths of alpha and beta are not equal we place an arrow on the edges directed toward the shorter root.
• In the noncrystallographic case, the edge connecting alpha and beta is labelled with the order of s_alpha s_beta (this is usually omitted if the order is 3).

  An  1---2---3- ... -n      Bn  1---2- ... -(n-1)=>=n

                                                     (n-1)
                                                    /
  Cn  1---2- ... -(n-1)=<=n    Dn  1---1- ... -(n-2)
                                                    \
                                                     n

  E6  1---3---4---5---6      E7  1---3---4---5---6---7
              |                          |
              2                          2

  E8  1---3---4---5---6---7---8
              |
              2

  F4  1---2=>=3---4          G2  1=<=2
                                   6

The only noncrystallographic root data are H_3, H_4 and I_2(m) for m = 5 and m > 6:

  H3  1---2---3
        5

  H4  1---2---3---4
        5

  I2(m) 1---2
          m

The labels on the vertices show the standard order used throughout the package, which is the same as that used in [Bou68]. Note that there is some redundancy in these diagrams. Specifically A_1=B_1=C_1=D_1, A_2=I_2(3), B_2=C_2=I_2(4), D_2=A_1 + A_1, D_3=A_3, G_2=I_2(6). The Dynkin diagram is completely determined by the Cartan matrix.

The Dynkin diagram does not determine the isomorphism type of a root datum and we say that two root data with the same Dynkin diagram have the same Cartan type.

The weights of a crystallographic root datum are those lambda in R Phi such that < lambda, alpha^star > in Z for every root alpha. The set of weights form a lattice Lambda called the weight lattice. We now have lattices Z Phi <= X <= Lambda (note that the second inclusion may not hold for nonsemisimple root data).

An isogeny from a root datum X_1, Phi_1, Y_1, Phi^ * _1) to a root datum (X_2, Phi_2, Y_2, Phi_2^ * ) is a homomorphism f : X_1 to X_2 whose image has finite index in X_2 and which induces a bijection between the root systems Phi_1 and Phi_2. Two root data are isogenous if they are related by isogenies to a third root datum. The isogeny type of the root datum within a given isogeny class is determined by the position of X between the root lattice Z Phi and the weight lattice Lambda. Alternatively, the isogeny type is determined by the isogeny group X/Z Phi within the fundamental group Lambda/Z Phi. We now list the fundamental groups for each irreducible Cartan type:

• A_n: Z/(n + 1)
• B_n, C_n, E_7: Z/2
• D_n: Z/4 for n odd; Z/2 x Z/2 for n even
• E_6: Z/3
• E_8, F_4, G_2: trivial.