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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 (Chapter COXETER 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.

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 α∈X and α^star∈Y, the linear map s_α:X -> X is defined by

x s_α= x - < x, α^star >α

and the linear map s_α^star:Y -> Y by

y s_α^star= y - < α, y >α^star.

These maps are called reflections if one of the following equivalent properties hold: < α, α^star >=2; (s_α)²=1; < xs_α, ys_α^star > = < x, y > for all x∈X and y∈Y; α s_α= - α. The mapping s_α^star is also called 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 Construction of Pseudo- reflections.

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 Φ is a finite subset of X - {0}. For each α in Φ, suppose a corresponding nonzero α^star in Y is given; set Φ^star={α^star | α∈Φ}. The tuple R=(X, Φ, Y, Φ^star) is called a root system if the following conditions are satisfied for every α in Φ

1.: s_α and s_α^star are reflections;
2.: Φ is closed under the action of s_α; and
3.: Φ^star is closed under the action of s_α^star.

The set X is called the root space and Y is called the coroot space. The elements of Φ are called roots and the elements of Φ^star are called coroots. A root system is said to be crystallographic if < α, β^star > is integral for every root α and coroot β^star. A root system is reduced, if α, β∈Φ with β a scalar product of α implies α=∓β. Note that it is possible for the set of roots to be empty, in which case the system is called toral.

Simple and Positive Roots

A subset Δ of Φ is called a set of simple roots if

1.: Δ is a basis for the span of the roots kΦ≤X; and
2.: Φ = Φ^ + ∪Φ^ -, where Φ^ + is the set of linear combinations of elements of Δ with nonnegative coefficients, and Φ^ - = - Φ^ +.

Every root system has a set of simple roots. Simple roots are frequently called fundamental roots. The elements of Φ^ + are called positive roots and the elements of Φ^ - are called negative roots. The coroots corresponding to the simple (respectively, positive, negative) roots are the simple (respectively, positive, negative) coroots.

The rank of a root system is the size of Δ, i.e. the dimension of the subspace kΦ. 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, the root system is said to be semisimple.

Choose a basis e₁, ..., e_d for X and a dual basis f₁, ..., f_d for Y, so that < e_i, f_j >=δ_ij. A reduced 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=< α_i, f_j > and B_ij=< e_j, α_i^star >.

The Coxeter Group

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

C = (< α_i, α_j^star >)_{i, j=1}ⁿ = AB^t.

Note that the root system is crystallographic if, and only if, its Cartan matrix is crystallographic. As in Chapter COXETER SYSTEMS, the Cartan matrix is used 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 reduced semisimple root systems. The isomorphism class of a reduced root system is determined by its Coxeter graph and its dimension.

A Coxeter form is a W-invariant bilinear form on X. If R is reduced and irreducible, then the roots can have at most two different lengths with respect to this form. We call the roots long or short accordingly. The Coxeter form is normalised so that the short roots in each 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).

Nonreduced Root Systems

A root system is reduced, if α, β∈Φ with β a scalar product of α implies α=∓β. A root α with the property 2α∉Φ is called reduced. A root α with the property (1/2)α∈Φ is called divisible. If R is a root system, then the set R₀ of indivisible roots in R form the indivisible subsystem.

Let R be a nonreduced irreducible crystallographic root system of rank n. It can be shown that R₀ is irreducible of type of type B_n and every root is either in R₀, or is two times a short root of R₀. The Cartan type of R in this case is BC_n. For noncrystallographic root systems the situation is more complex.

Note that the Cartan matrix, Coxeter matrix, Coxeter diagram, Coxeter group and Dynkin diagram are the same for R and R₀. Thus, when creating a non-reduced crystallographic root system for a given Cartan matrix, Coxeter matrix, Coxeter diagram, Coxeter group or Dynkin diagram, one must specify the set of nonreduced simple roots. For example, let C be a cartan matrix of type B₂ x B₃. Then the set of non-reduced fundamental roots can be one of emptyset, {2}, {5}, or {2, 5}, in which cases the root system will be of types B₂ x B₃, BC₂ x B₃, B₂ x BC₃, or BC₂ x BC₃ respectively.