# 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](../ReflectionGroups/index-reflection-groups.md#chapgrprfl)). Root systems are essential in the theories of finite Coxeter groups (Chapter [Coxeter Groups](../CoxeterGroups/index-coxeter-groups.md#chapgrpcox)) and Lie algebras (Chapter [Lie Algebras](../LieAlgebras/index-lie-algebras.md#chapalglie)). See [[Bourbaki, 1968](../../references.md#cite-bourbakilie)] for more details on the theory of root systems. The closely related concept of a root datum is discussed in Chapter [Root Data](../RootData/index-root-data.md#chaprootdtm).

## Reflections

Let $X$ and $Y$ be vector spaces over a field $k$ with bilinear pairing $\langle\circ,\circ\rangle:X\times Y\rightarrow k$ that identifies $Y$ with the dual of $X$. Given nonzero $\alpha\in X$ and $\alpha^\star\in Y$, the linear map $s_\alpha:X\rightarrow X$ is defined by

$$
x s_\alpha= x - \langle x,\alpha^\star\rangle\alpha
$$

and the linear map $s_\alpha^\star:Y\rightarrow Y$ by

$$
y s_\alpha^\star= y - \langle\alpha,y \rangle\alpha^\star.
$$

These maps are called *reflections* if one of the following equivalent properties hold: $\langle\alpha,\alpha^\star\rangle=2$; ${s_\alpha}^2=1$; $\langle xs_\alpha,ys_\alpha^\star\rangle = \langle x,y\rangle$ for all $x\in X$ and $y\in Y$; $\alpha s_\alpha=-\alpha$. The mapping $s_\alpha^\star$ 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](../ReflectionGroups/reflections.md#sectrefgrprfl).

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 $\langle x,y\rangle = xy^T$. 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](../../BasicRings/Rationals/index-rationals.md#fldrat)), a number field (Chapter [Rational Field](../../BasicRings/Rationals/index-rationals.md#fldrat)), or a cyclotomic field (Chapter [Cyclotomic Fields](../../GlobalFields/CyclotomicFields/index-cyclotomic-fields.md#chapfldcyc)). The real field (Chapter [Real and Complex Fields](../../BasicRings/RealAndComplexFields/index-real-and-complex-fields.md#chapfldre)) is *not* allowed since it is not infinite precision.

## Definition of a Root System

Suppose $\Phi$ is a finite subset of $X\setminus \{0\}$. For each $\alpha$ in $\Phi$, suppose a corresponding nonzero $\alpha^\star$ in $Y$ is given; set $\Phi^\star=\{\alpha^\star\mid\alpha\in\Phi\}$. The tuple $R=(X,\Phi,Y,\Phi^\star)$ is called 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 $s_\alpha^\star$.

The set $X$ is called the *root space* and $Y$ is called the *coroot space*. The elements of $\Phi$ are called *roots* and the elements of $\Phi^\star$ are called *coroots*. A root system is said to be *crystallographic* if $\langle\alpha,\beta^\star\rangle$ is integral for every root $\alpha$ and coroot $\beta^\star$. A root system is *reduced*, if $\alpha,\beta\in\Phi$ with $\beta$ a scalar product of $\alpha$ implies $\alpha=\pm\beta$. 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 $\Delta$ of $\Phi$ is called a set of *simple roots* if

1. $\Delta$ is a basis for the span of the roots $k\Phi\le X$; and

2. $\Phi = \Phi^+\cup\Phi^-$, where $\Phi^+$ is the set of linear combinations of elements of $\Delta$ with nonnegative coefficients, and $\Phi^- = -\Phi^+$. Every root system has a set of simple roots. Simple roots are frequently called fundamental roots. The elements of $\Phi^+$ are called *positive roots* and the elements of $\Phi^-$ 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 $\Delta$, 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, the root system is said to be *semisimple*.

Choose a basis $e_1,\dots,e_d$ for $X$ and a dual basis $f_1,\dots,f_d$ for $Y$, so that $\langle e_i,f_j\rangle=\delta_{ij}$. 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. $A_{ij}=\langle\alpha_i,f_j\rangle$ and $B_{ij}=\langle e_j,\alpha_i^\star\rangle$.

## 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

$$
C =
  \left(\left\langle\,\alpha_i,\alpha_j^\star\,\right\rangle\right)_{i,j=1}^n =
  AB^t.
$$

Note that the root system is crystallographic if, and only if, its Cartan matrix is crystallographic. As in Chapter [Coxeter Systems](../CoxeterSystems/index-coxeter-systems.md#chapcartan), 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](../CoxeterSystems/finiteaffine.md#sectcartanfinaff) 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 $\langle\circ,\circ\rangle$; however it can be arranged for them to be the same (see [`StandardRootSystem`](construct-root-system.md#function-intrstandardrootsystem)).

## Nonreduced Root Systems

A root system is *reduced*, if $\alpha,\beta\in\Phi$ with $\beta$ a scalar product of $\alpha$ implies $\alpha=\pm\beta$. A root $\alpha$ with the property $2\alpha\notin \Phi$ is called *reduced*. A root $\alpha$ with the property ${1\over 2}\alpha\in\Phi$ is called *divisible*. If $R$ is a root system, then the set $R_0$ 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 $R_0$ is irreducible of type of type $B_n$ and every root is either in $R_0$, or is two times a short root of $R_0$. 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 $R_0$. 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 $B_2\times B_3$. 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 $B_2\times B_3$, $BC_2\times B_3$, $B_2\times BC_3$, or $BC_2\times BC_3$ respectively.