Introduction

This chapter describes Magma functions for computing with finite real root systems. A root system describes the reflections in a reflection group (Chapter REFLECTION GROUPS). Root systems are essential in the theories of finite Coxeter groups (Chapters COXETER GROUPS and COXETER GROUPS AS PERMUTATION GROUPS) and Lie algebras (Chapter LIE ALGEBRAS). See [Bou68] for more details on the theory of root systems. The closely related concept of a root datum is discussed in Chapter ROOT DATA.

Prior to Version 2.10 of magma, we did not make the distinction between root systems and root data. If you are working with Lie algebras or groups of Lie type, root data can be used as before. If you are working with Coxeter groups or reflection groups, you are probably only interested in root systems.

The original release of this package was modelled on part of the Chevie package of GAP [GHL+96] by Meinholf Geck, Frank Lübeck, Jean Michel and Götz Pfeiffer.

Subsections

• Reflections
• Definition of a Root System
• Simple and Positive Roots
• The Coxeter Group

Reflections

Let X and Y be vector spaces over a field k with bilinear pairing < , >:X x Y -> k that identifies Y with the dual of X. Given nonzero alpha in X and alpha^star in Y, we define the linear map s_alpha:X -> X by

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

and 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 also call s_alpha^star a coreflection: this just means it is a reflection defined on Y instead of X. magma functions for computing with reflections are described in Section Constructing Reflections and Pseudoreflections.

If X has an inner product, then we can take Y=X and use the inner product as our pairing. In magma, we generally take X=Y to be a row space, with the bilinear pairing given by the standard inner product < x, y > = xy^T. However, it is sometimes useful to allow X and Y to be distinct subspaces of a row space.

For the purposes of this chapter, k will always be the rational field (Chapter RATIONAL FIELD), a number field (Chapter RATIONAL FIELD), or a cyclotomic field (Chapter CYCLOTOMIC FIELDS); the real field (Chapter REAL AND COMPLEX FIELDS) is not allowed since it is not infinite precision.

Definition of a Root System

Suppose Phi is a finite subset of X - {0}. For each alpha in Phi, suppose we have a corresponding nonzero alpha^star in Y; set Phi^star={alpha^star | alpha in Phi}. We say that R=(X, Phi, Y, Phi^star) is a root system if the following conditions are satisfied for every alpha in Phi

1.

s_alpha and s_alpha^star are reflections;

2.

Phi is closed under the action of s_alpha; and

3.

Phi^star is closed under the action of s_alpha^star.

We call X the root space and Y the coroot space. The elements of Phi are called roots and the elements of Phi^star are called coroots. We call a root system crystallographic if < alpha, beta^star > is integral for every root alpha and coroot beta^star.

Simple and Positive Roots

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

1.

Delta is a basis for the span of the roots k Phi <= X; and

2.

Phi = Phi^ + union Phi^ -, where Phi^ + is the set of linear combinations of elements of Delta with nonnegative coefficients, and Phi^ - = - Phi^ +.

The rank of a root system is the size of Delta, i.e. the dimension of the subspace k Phi. The rank cannot be larger than the dimension of the root system (i.e. the dimension of X); if the rank and dimension are equal, we say the root system is semisimple.

Choose a basis e_1, ..., e_d for X and a dual basis f_1, ..., f_d for Y, so that < e_i, f_j >=delta_(ij). The root system is determined by a pair of real matrices A and B where the rows of A are the simple roots and the rows of B are the corresponding coroots; i.e. A_(ij)=< alpha_i, f_j > and B_(ij)=< e_j, alpha_i^star >.

The Coxeter Group

The group W generated by the reflections s_alpha, for alpha a simple root, is a finite Coxeter group. The Cartan matrix of a root system is

C = (< alpha_i, alpha_j^star >)_(i, j=1)^n = AB^t.

Note that the root system is crystallographic if, and only if, its Cartan matrix is. As in Chapter COXETER SYSTEMS, we use the Cartan matrix to define the Coxeter matrix, Coxeter graph, and Dynkin digraph of a root system.

The classification of Section Finite and Affine Coxeter Groups applies to semisimple root systems. The isomorphism class of an arbitrary root system is determined by its Coxeter graph and its dimension.

The Coxeter form

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

is a W-invariant bilinear form on X. We normalise the Coxeter form so that the short roots in each crystallographic component have length one. Note that, even if X=Y, this form will generally not be the same as the pairing < , >; however it can be arranged for them to be the same (see StandardRootSystem).