# Introduction

This chapter describes Magma functions for computing with (extended) root data. Root data are fundamental to Lie theory: Lie algebras (Chapter [Lie Algebras](../LieAlgebras/index-lie-algebras.md#chapalglie)) and groups of Lie type (Chapter [Groups of Lie Type](../ChevalleyGroups/index-chevalley-groups.md#chapgrplie)). Our description of split reduced root data follows [[Demazure, 1970](../../references.md#cite-demazure)] and [[Carter, 1993](../../references.md#cite-carter-big)] except that reflections act on the right as in customary in Magma. Our description of extended root data follows [[Satake, 1971](../../references.md#cite-satake-class)], [[Schattschneider, 1969](../../references.md#cite-schattschneider)], and [[Haller, 2005](../../references.md#cite-sh)]. Our description of split non-reduced root data follows [[Bourbaki, 1968](../../references.md#cite-bourbakilie)].

The closely related concept of a root system is discussed in Chapter [Root Systems](../RootSystems/index-root-systems.md#chaprootsys). When working with Lie algebras or groups of Lie type, root data should be used. When working with Coxeter groups (Chapter [Coxeter Groups](../CoxeterGroups/index-coxeter-groups.md#chapgrpcox)) or reflection groups (Chapter [Reflection Groups](../ReflectionGroups/index-reflection-groups.md#chapgrprfl)), it is likely that only root systems are of interest.

## Reflections

Let $X$ and $Y$ be free ${\mathbb{Z}}$-modules with bilinear pairing $\langle\circ,\circ\rangle:X\times Y\rightarrow {\mathbb{Z}}$ that identifies $Y$ with the dual of $X$. Given nonzero $\alpha\in X$ and $\alpha^\star\in Y$, we define the ${\mathbb{Z}}$-linear map $s_\alpha:X\rightarrow X$ by

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

and the ${\mathbb{Z}}$-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 map $s_\alpha^\star$ 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, $X$ and $Y$ are usually standard ${\mathbb{Z}}$-modules. However, it is sometimes useful to allow $X$ and $Y$ to be distinct sublattices of a standard lattice. The bilinear pairing is always given by the standard inner product: $\langle x,y\rangle = xy^T$.

## Definition of a Split Root Datum

Suppose $\Phi$ is a finite subset of $X\setminus \{0\}$. For each $\alpha$ in $\Phi$, suppose there is a corresponding $\alpha^\star$ in $Y\setminus \{0\}$; set $\Phi^\star=\{\alpha^\star\mid\alpha\in\Phi\}$. The datum $R=(X,\Phi,Y,\Phi^\star)$ is said to be a *(split) root datum* 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 lattice $X$ is called the *full root lattice* and $Y$ the *full coroot lattice*. The vector space $X\otimes{\mathbb{Q}}$ is called the *root space* and $Y\otimes{\mathbb{Q}}$ the *coroot space*. The elements of $\Phi$ are called *roots* and the elements of $\Phi^\star$ are called *coroots*. A root datum is *reduced*, if $\alpha,\beta\in\Phi$ with $\beta$ a scalar product of $\alpha$ implies $\alpha=\pm\beta$.

## Simple and Positive Roots

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

1. $\Delta$ is a basis for the rational span of the roots ${\mathbb{Q}}\Phi\le {\mathbb{Q}}\otimes X$; 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 datum 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 $\Phi^-$ *negative roots*. The coroots corresponding to the simple (resp. positive, negative) roots are the *simple* (respectively, *positive*, *negative*) *coroots*.

The *rank* of the root datum is the size of $\Delta$, i.e. the dimension of the subspace ${\mathbb{Q}}\Phi$. The rank cannot be larger than the *dimension* of the root datum (i.e. the dimension of ${\mathbb{Q}}\otimes X$). If the rank and dimension are equal, the root datum 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 integral 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 datum is

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

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 datum.

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 often be arranged for them to be the same (see [`StandardRootSystem`](../RootSystems/construct-root-system.md#function-intrstandardrootsystem)).

## Nonreduced Root Data

A root datum 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 datum, then the set $R_0$ of indivisible roots in $R$ form the *indivisible subsystem*.

Let $R$ be a nonreduced irreducible root datum 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 $BC_n$.

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 root datum for a given Cartan matrix, Coxeter matrix, Coxeter diagram, Coxeter group or Dynkin diagram, one must specify the set of non-reduced fundamental roots. E.g., let $C$ be a cartan matrix of type $B_2\times B_3$. Then the set of nonreduced fundamental roots can be one of $\emptyset$, $\{2\}$, $\{5\}$ or $\{2,5\}$, in which cases the root datum 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.

## Isogeny of Split Reduced Root Data

The Dynkin digraph and dimension do not completely determine the isomorphism type of a split root datum, as the Coxeter graph and dimension do for a root system. Two root data with isomorphic Dynkin digraphs are said to be *Cartan equivalent*. We now describe the isomorphism classes within each Cartan equivalence class of split reduced irreducible root data. Since every semisimple reduced root datum is isogenous to a direct sum of irreducible root data, this immediately gives a classification of the split semisimple root data. Classifying nonsemisimple root data would be more complicated.

The *weights* of a root datum are the $\lambda$ in ${\mathbb{Q}}\Phi\le X\otimes{\mathbb{Q}}$ such that $\langle\lambda,\alpha^\star\rangle\in{\mathbb{Z}}$ for every coroot $\alpha^\star$. The weights form a lattice $\Lambda$ called the *weight lattice*. We now have lattices ${\mathbb{Z}}\Phi\le X \le\Lambda$ (note that the second inclusion holds only for semisimple root data). The isomorphism class of a root datum in a fixed Cartan equivalence class is determined by the position of $X$ between the root lattice ${\mathbb{Z}}\Phi$ and the weight lattice $\Lambda$. Alternatively, the isomorphism class is determined by the *isogeny group* $X/{\mathbb{Z}}\Phi$ within the *fundamental group* $\Lambda/{\mathbb{Z}}\Phi$. The fundamental group is determined by the Cartan matrix $C$: it is isometric to ${\mathbb{Z}}^n/\Theta$ where $\Theta$ is the lattice generated by the rows of $C$. The fundamental groups of the irreducible Cartan equivalence classes are

**•**
$A_n$: ${\mathbb{Z}}/(n+1)$;

**•**
$B_n$, $C_n$, $E_7$: ${\mathbb{Z}}/2$;

**•**
$D_n$: ${\mathbb{Z}}/4$ for $n$ odd, ${\mathbb{Z}}/2 \times {\mathbb{Z}}/2$ for $n$ even;

**•**
$E_6$: ${\mathbb{Z}}/3$;

**•**
$E_8$, $F_4$, $G_2$: trivial. If $X={\mathbb{Z}}\Phi$ the root datum is said to be *adjoint*; if $X=\Lambda$ it is said to be *simply connected*. The quotient $Y/{\mathbb{Z}}\Phi^\star$ is called the *coisogeny group*; in the semisimple case it is isomorphic to $\Lambda/{\mathbb{Z}}\Phi$.

## Extended Root Data

An extended root datum is a split root datum $R=(X,\Phi,Y,\Phi^\star)$ and a permutation group $\Gamma$ with actions on $X$ and $Y$ that respect the pairing $\langle\circ,\circ\rangle$.

Fix a set of simple roots $\Delta$. Let $O(\chi)$ denote the orbit of $\chi\in X$ under the $\Gamma$-action. Then, for $\alpha\in\Phi$ either $O(\alpha)$ is contained in $\Phi^+$, or it is contained in $\Phi^-$, or the sum of the roots of $O(\alpha)$ is zero. We call $O(\alpha)$ a *positive*, *negative* or *zero orbit*, respectively. Put

$$
X_0      := \{ \chi \in X \mid
          \sum_{\gamma\in\Gamma} \chi^{\gamma} = 0 \}.
$$

Let $\Phi_0 := \Phi \cap X_0$ and $\Delta_0 := \Delta \cap X_0$. Then $X_0$ is a submodule of $X$, $\Phi_0$ is a subsystem of $\Phi$, and $\Delta_0$ is a fundamental system of $\Phi_0$. Note that $\Delta_0$ is not necessarily a basis of $X_0$. Analogously, we define $Y_0$ and $\Phi^\star_0$. The subdatum $R_0 = (X_0, \Phi_0, Y_0, \Phi^\star_0)$ is called the *anisotropic subdatum* of $R$.

Set $\bar{X} := X/X_0$ and let $\pi:X\to \bar{X}$ be the standard projection. Then $\bar{X}$ is a free ${\mathbb{Z}}$-module and $\pi$ is a homomorphism of modules. Let $\bar{\Phi}$ and $\bar{\Delta}$ be the images under $\pi$ of $\Phi \setminus \Phi_0$ and $\Delta \setminus \Delta_0$, respectively. Then $\bar{\Phi}$ is a root system and $\bar{\Delta}$ is a fundamental system of it. We call $\bar{\Phi}$ the *relative root system* and $\bar{\Delta}$ the *relative fundamental system*. Note that $\bar{\Phi}$ need not be irreducible nor reduced even if $\Phi$ is. The rank of the relative system is $|\bar{\Delta}|$ and is called the *relative rank*, whereas the rank $|\Delta|$ of $\Phi$ is called the *absolute rank*. Let $\bar{\Phi}^+$ and $\bar{\Phi}^-$ denote the images under $\pi$ of $\Phi^+ \setminus \Phi_0$ and $\Phi^- \setminus \Phi_0$. When $X_0=X$, the relative root system is an empty set and the form is called *anisotropic*.

Each $\gamma\in\Gamma$ acts on $X$ by $\chi \mapsto \chi^{\sigma w}$ for some unique $w\in W$ and $\sigma$ a Dynkin diagram symmetry. By $\alpha \mapsto \alpha^\sigma$ for $\alpha\in \Delta$ we define the $[\Gamma]$-action on $\Delta$. The extended root datum is called *inner* if the $[\Gamma]$-action is trivial and *outer* otherwise. The orbits of the $[\Gamma]$-action, that are not contained in $X_0$ are called *distinguished*.

An extended root datum is called *twisted* if the $\Gamma$-action is not trivial.

The (split) Cartan name of an extended root datum is the name of the corresponding split root datum. An extended root datum is *absolutely irreducible* if the corresponding split datum is irreducible. It is irreducible if there is no direct sum decomposition of the split datum which is preserved under the action of $\Gamma$. The *twisted Cartan name* of a root datum is the Cartan name, with extra information describing the twist. The name ${}^mX_{n,e}$ indicates a root datum with split Cartan name $X_n$, where the kernel of the $[\Gamma]$-action has index $m$ in $\Gamma$, and $e$ is the rank of the relative root system. The twisted Cartan name describes absolutely irreducible root data up to isomorphism. This is not true for simple root data however.