Introduction

Introduction#

This chapter describes Magma functions for computing with (extended) root data. Root data are fundamental to Lie theory: Lie algebras (Chapter ChapAlgLie) and groups of Lie type (Chapter ChapGrpLie). Our description of split reduced root data follows [Demazure, 1970] and [Carter, 1993] except that reflections act on the right as in customary in . Our description of extended root data follows [Satake, 1971], [Schattschneider, 1969], and [Haller, 2005]. Our description of split non-reduced root data follows [Bourbaki, 1968].

The closely related concept of a root system is discussed in Chapter ChapRootSys. When working with Lie algebras or groups of Lie type, root data should be used. When working with Coxeter groups (Chapter ChapGrpCox) or reflection groups (Chapter 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 \(X\). “ 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 , \(X\) 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 \(\Phi\)

\(s_\alpha\) and \(s_\alpha^\star\) are reflections;
\(\Phi\) is closed under the action of \(s_\alpha\); and
\(\Phi^\star\) is closed under the action of \(s_\alpha^\star\).

The lattice \(X\) is called the full root lattice and \(Y\) the full coroot lattice. The vector space \(X\tens{\mathbb{Q}}\) is called the root space and \(Y\tens{\mathbb{Q}}\) 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 \({\mathbb{Q}}\Phi\le {\mathbb{Q}}\tens X\); 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 \({\mathbb{Q}}\tens X\)). 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 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 ).

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 \({\mathbb{Q}}\Phi\le X\tens{\mathbb{Q}}\) 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.