.. _SectRDIntro: .. _introduction: Introduction ============ This chapter describes Magma functions for computing with (extended) root data. Root data are fundamental to Lie theory: Lie algebras (Chapter :ref:`ChapAlgLie`) and groups of Lie type (Chapter :ref:`ChapGrpLie`). Our description of split reduced root data follows :cite:`Demazure` and :cite:`Carter-big` except that reflections act on the right as in customary in . Our description of extended root data follows :cite:`satake_class`, :cite:`Schattschneider`, and :cite:`SH`. Our description of split non-reduced root data follows :cite:`BourbakiLie`. The closely related concept of a root system is discussed in Chapter :ref:`ChapRootSys`. When working with Lie algebras or groups of Lie type, root data should be used. When working with Coxeter groups (Chapter :ref:`ChapGrpCox`) or reflection groups (Chapter :ref:`ChapGrpRfl`), it is likely that only root systems are of interest. .. _SubsectRSRfl: .. _refl: Reflections ----------- Let :math:`X` and :math:`Y` be free :math:`{\mathbb{Z}}`-modules with bilinear pairing :math:`\langle\circ,\circ\rangle:X\times Y\rightarrow {\mathbb{Z}}` that identifies :math:`Y` with the dual of :math:`X`. Given nonzero :math:`\alpha\in X` and :math:`\alpha^\star\in Y`, we define the :math:`{\mathbb{Z}}`-linear map :math:`s_\alpha:X\rightarrow X` by .. math:: x s_\alpha= x - \langle x,\alpha^\star\rangle\alpha and the :math:`{\mathbb{Z}}`-linear map :math:`s_\alpha^\star:Y\rightarrow Y` by .. math:: y s_\alpha^\star= y - \langle\alpha,y \rangle\alpha^\star. These maps are called *reflections* if one of the following equivalent properties hold: :math:`\langle\alpha,\alpha^\star\rangle=2`; :math:`{s_\alpha}^2=1`; :math:`\langle xs_\alpha,ys_\alpha^\star\rangle = \langle x,y\rangle` for all :math:`x\in X` and :math:`y\in Y`; :math:`\alpha s_\alpha=-\alpha`. The map :math:`s_\alpha^\star` is also called a *coreflection*: this just means it is a reflection defined on :math:`Y` instead of :math:`X`. “ functions for computing with reflections are described in Section :ref:`SectRefGrpRfl`. If :math:`X` has an inner product, then we can take :math:`Y=X` and use the inner product as our pairing. In , :math:`X` and :math:`Y` are usually standard :math:`{\mathbb{Z}}`-modules. However, it is sometimes useful to allow :math:`X` and :math:`Y` to be distinct sublattices of a standard lattice. The bilinear pairing is always given by the standard inner product: :math:`\langle x,y\rangle = xy^T`. .. _SubsectRDDefn: .. _rootdtm: Definition of a Split Root Datum -------------------------------- Suppose :math:`\Phi` is a finite subset of :math:`X\setminus \{0\}`. For each :math:`\alpha` in :math:`\Phi`, suppose there is a corresponding :math:`\alpha^\star` in :math:`Y\setminus \{0\}`; set :math:`\Phi^\star=\{\alpha^\star\mid\alpha\in\Phi\}`. The datum :math:`R=(X,\Phi,Y,\Phi^\star)` is said to be a *(split) root datum* if the following conditions are satisfied for every :math:`\alpha` in :math:`\Phi` 1. :math:`s_\alpha` and :math:`s_\alpha^\star` are reflections; 2. :math:`\Phi` is closed under the action of :math:`s_\alpha`; and 3. :math:`\Phi^\star` is closed under the action of :math:`s_\alpha^\star`. The lattice :math:`X` is called the *full root lattice* and :math:`Y` the *full coroot lattice*. The vector space :math:`X\tens{\mathbb{Q}}` is called the *root space* and :math:`Y\tens{\mathbb{Q}}` the *coroot space*. The elements of :math:`\Phi` are called *roots* and the elements of :math:`\Phi^\star` are called *coroots*. A root datum is *reduced*, if :math:`\alpha,\beta\in\Phi` with :math:`\beta` a scalar product of :math:`\alpha` implies :math:`\alpha=\pm\beta`. .. _SubsectRDPosSimple: .. _positive-simple-roots: Simple and Positive Roots ------------------------- A subset :math:`\Delta` of :math:`\Phi` is called a set of *simple roots* if 1. :math:`\Delta` is a basis for the rational span of the roots :math:`{\mathbb{Q}}\Phi\le {\mathbb{Q}}\tens X`; and 2. :math:`\Phi = \Phi^+\cup\Phi^-`, where :math:`\Phi^+` is the set of linear combinations of elements of :math:`\Delta` with nonnegative coefficients, and :math:`\Phi^- = -\Phi^+`. Every root datum has a set of simple roots. Simple roots are frequently called fundamental roots. The elements of :math:`\Phi^+` are called *positive roots* and the elements of :math:`\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 :math:`\Delta`, i.e. the dimension of the subspace :math:`{\mathbb{Q}}\Phi`. The rank cannot be larger than the *dimension* of the root datum (i.e. the dimension of :math:`{\mathbb{Q}}\tens X`). If the rank and dimension are equal, the root datum is said to be *semisimple*. Choose a basis :math:`e_1,\dots,e_d` for :math:`X` and a dual basis :math:`f_1,\dots,f_d` for :math:`Y`, so that :math:`\langle e_i,f_j\rangle=\delta_{ij}`. A reduced root system is determined by a pair of integral matrices :math:`A` and :math:`B` where the rows of :math:`A` are the simple roots and the rows of :math:`B` are the corresponding coroots; i.e. :math:`A_{ij}=\langle\alpha_i,f_j\rangle` and :math:`B_{ij}=\langle e_j,\alpha_i^\star\rangle`. .. _SubsectRDCoxGrp: .. _rootdatum-cox-grps: The Coxeter Group ----------------- The group :math:`W` generated by the reflections :math:`s_\alpha`, for :math:`\alpha` a simple root, is a finite Coxeter group. The *Cartan matrix* of a root datum is .. math:: C = \left(\left\langle\,\alpha_i,\alpha_j^\star\,\right\rangle\right)_{i,j=1}^n = AB^t. As in Chapter :ref:`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 :math:`W`-invariant bilinear form on :math:`X`. If :math:`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 :math:`X=Y`, this form will generally not be the same as the pairing :math:`\langle\circ,\circ\rangle`; however it can often be arranged for them to be the same (see ). .. _SubsectRDNonRed: .. _nonred-root-data: Nonreduced Root Data -------------------- A root datum is *reduced*, if :math:`\alpha,\beta\in\Phi` with :math:`\beta` a scalar product of :math:`\alpha` implies :math:`\alpha=\pm\beta`. A root :math:`\alpha` with the property :math:`2\alpha\notin \Phi` is called *reduced*. A root :math:`\alpha` with the property :math:`{1\over 2}\alpha\in\Phi` is called *divisible*. If :math:`R` is a root datum, then the set :math:`R_0` of indivisible roots in :math:`R` form the *indivisible subsystem*. Let :math:`R` be a nonreduced irreducible root datum of rank :math:`n`. It can be shown that :math:`R_0` is irreducible of type of type :math:`B_n` and every root is either in :math:`R_0`, or is two times a short root of :math:`R_0`. The Cartan type of :math:`R` in this case is :math:`BC_n`. Note that the Cartan matrix, Coxeter matrix, Coxeter diagram, Coxeter group and Dynkin diagram are the same for :math:`R` and :math:`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 :math:`C` be a cartan matrix of type :math:`B_2\times B_3`. Then the set of nonreduced fundamental roots can be one of :math:`\emptyset`, :math:`\{2\}`, :math:`\{5\}` or :math:`\{2,5\}`, in which cases the root datum will be of types :math:`B_2\times B_3`, :math:`BC_2\times B_3`, :math:`B_2\times BC_3` or :math:`BC_2\times BC_3` respectively. .. _SubsectRDIsogeny: .. _rootdata: 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 :math:`\lambda` in :math:`{\mathbb{Q}}\Phi\le X\tens{\mathbb{Q}}` such that :math:`\langle\lambda,\alpha^\star\rangle\in{\mathbb{Z}}` for every coroot :math:`\alpha^\star`. The weights form a lattice :math:`\Lambda` called the *weight lattice*. We now have lattices :math:`{\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 :math:`X` between the root lattice :math:`{\mathbb{Z}}\Phi` and the weight lattice :math:`\Lambda`. Alternatively, the isomorphism class is determined by the *isogeny group* :math:`X/{\mathbb{Z}}\Phi` within the *fundamental group* :math:`\Lambda/{\mathbb{Z}}\Phi`. The fundamental group is determined by the Cartan matrix :math:`C`: it is isometric to :math:`{\mathbb{Z}}^n/\Theta` where :math:`\Theta` is the lattice generated by the rows of :math:`C`. The fundamental groups of the irreducible Cartan equivalence classes are :math:`A_n`: :math:`{\mathbb{Z}}/(n+1)`; :math:`B_n`, :math:`C_n`, :math:`E_7`: :math:`{\mathbb{Z}}/2`; :math:`D_n`: :math:`{\mathbb{Z}}/4` for :math:`n` odd, :math:`{\mathbb{Z}}/2 \times {\mathbb{Z}}/2` for :math:`n` even; :math:`E_6`: :math:`{\mathbb{Z}}/3`; :math:`E_8`, :math:`F_4`, :math:`G_2`: trivial. If :math:`X={\mathbb{Z}}\Phi` the root datum is said to be *adjoint*; if :math:`X=\Lambda` it is said to be *simply connected*. The quotient :math:`Y/{\mathbb{Z}}\Phi^\star` is called the *coisogeny group*; in the semisimple case it is isomorphic to :math:`\Lambda/{\mathbb{Z}}\Phi`. .. _SubsectExtRD: .. _extended-rootdtm: Extended Root Data ------------------ An extended root datum is a split root datum :math:`R=(X,\Phi,Y,\Phi^\star)` and a permutation group :math:`\Gamma` with actions on :math:`X` and :math:`Y` that respect the pairing :math:`\langle\circ,\circ\rangle`. Fix a set of simple roots :math:`\Delta`. Let :math:`O(\chi)` denote the orbit of :math:`\chi\in X` under the :math:`\Gamma`-action. Then, for :math:`\alpha\in\Phi` either :math:`O(\alpha)` is contained in :math:`\Phi^+`, or it is contained in :math:`\Phi^-`, or the sum of the roots of :math:`O(\alpha)` is zero. We call :math:`O(\alpha)` a *positive*, *negative* or *zero orbit*, respectively. Put .. math:: X_0 := \{ \chi \in X \mid \sum_{\gamma\in\Gamma} \chi^{\gamma} = 0 \}. Let :math:`\Phi_0 := \Phi \cap X_0` and :math:`\Delta_0 := \Delta \cap X_0`. Then :math:`X_0` is a submodule of :math:`X`, :math:`\Phi_0` is a subsystem of :math:`\Phi`, and :math:`\Delta_0` is a fundamental system of :math:`\Phi_0`. Note that :math:`\Delta_0` is not necessarily a basis of :math:`X_0`. Analogously, we define :math:`Y_0` and :math:`\Phi^\star_0`. The subdatum :math:`R_0 = (X_0, \Phi_0, Y_0, \Phi^\star_0)` is called the *anisotropic subdatum* of :math:`R`. Set :math:`\bar{X} := X/X_0` and let :math:`\pi:X\to \bar{X}` be the standard projection. Then :math:`\bar{X}` is a free :math:`{\mathbb{Z}}`-module and :math:`\pi` is a homomorphism of modules. Let :math:`\bar{\Phi}` and :math:`\bar{\Delta}` be the images under :math:`\pi` of :math:`\Phi \setminus \Phi_0` and :math:`\Delta \setminus \Delta_0`, respectively. Then :math:`\bar{\Phi}` is a root system and :math:`\bar{\Delta}` is a fundamental system of it. We call :math:`\bar{\Phi}` the *relative root system* and :math:`\bar{\Delta}` the *relative fundamental system*. Note that :math:`\bar{\Phi}` need not be irreducible nor reduced even if :math:`\Phi` is. The rank of the relative system is :math:`|\bar{\Delta}|` and is called the *relative rank* , whereas the rank :math:`|\Delta|` of :math:`\Phi` is called the *absolute rank*. Let :math:`\bar{\Phi}^+` and :math:`\bar{\Phi}^-` denote the images under :math:`\pi` of :math:`\Phi^+ \setminus \Phi_0` and :math:`\Phi^- \setminus \Phi_0`. When :math:`X_0=X`, the relative root system is an empty set and the form is called *anisotropic*. Each :math:`\gamma\in\Gamma` acts on :math:`X` by :math:`\chi \mapsto \chi^{\sigma w}` for some unique :math:`w\in W` and :math:`\sigma` a Dynkin diagram symmetry. By :math:`\alpha \mapsto \alpha^\sigma` for :math:`\alpha\in \Delta` we define the :math:`[\Gamma]`-action on :math:`\Delta`. The extended root datum is called *inner* if the :math:`[\Gamma]`-action is trivial and *outer* otherwise. The orbits of the :math:`[\Gamma]`-action, that are not contained in :math:`X_0` are called *distinguished*. An extended root datum is called *twisted* if the :math:`\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 :math:`\Gamma`. The *twisted Cartan name* of a root datum is the Cartan name, with extra information describing the twist. The name :math:`{}^mX_{n,e}` indicates a root datum with split Cartan name :math:`X_n`, where the kernel of the :math:`[\Gamma]`-action has index :math:`m` in :math:`\Gamma`, and :math:`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.