.. _SectGrpLieIntro:

.. _introduction:

Introduction
============

This chapter describes Magma functions for computing with groups of Lie type.
These functions are based on :cite:`CohenMurrayTaylor` for split
types, and :cite:`SH` for twisted types.

Given an extended root datum and ring with a :math:`\Gamma`-action, a group of
Lie type can be constructed in Magma. Such groups include reductive Lie groups
(when the ring is or ), reductive algebraic groups (when the ring is an
algebraically closed field), and finite groups of Lie type (when the ring is a
finite field).

.. _SubsectGrpLieSteinberg:

.. _steinberg:

The Steinberg Presentation
--------------------------

The approach to computation in split groups of Lie type described here is based
on the Steinberg presentation:cite:`Steinberg-presentation` Let
:math:`G` be a split group of Lie type with root datum :math:`R` over the ring
:math:`k`. Suppose the roots of :math:`R` are :math:`\alpha_1,\dots,\alpha_{2N}`
ordered as in Section :ref:`SectRDRoot` and :math:`n` is the rank
of :math:`R`. Then :math:`G` contains *root elements*
:math:`x_r(t)=x_{\alpha_r}(t)` for :math:`t` in :math:`k`. If :math:`R` is
semisimple, the root elements generate :math:`G`. In the general case, it is
necessary to introduce extra torus elements. Let :math:`Y={\mathbb{Z}}^d` be the
coroot space of the root datum. The torus is taken to be the abelian group
:math:`Y\otimes k^\times`, represented as the set of vectors in :math:`k^d` with
each component invertible, and multiplication is performed componentwise. The
Weyl group of :math:`G` is just the Coxeter group of the root datum :math:`R`.
Redundant generators :math:`n_r` are also included, corresponding to the
generators :math:`s_r` of the Weyl group.

Since the generating set is parametrised by field elements it is generally not
possible to define :math:`G` within the category of finitely presented groups
``GrpFP``, so groups of Lie type form their own category, ``GrpLie``.

Note that groups of Lie type in Magma are designed primarily for fields whose
elements are exact. While it is possible to define these groups over real and
complex fields (Chapter :ref:`ChapFldRe`), no attempt has been made
to control rounding error in this case.

.. _SubsectGrpLieBruhat:

.. _bruhat:

Bruhat Normalisation
--------------------

The Bruhat decomposition :raw-latex:`\cite[Chapter
2]{Carter-big}` gives us a useful normal form for elements of a split group of
Lie type defined over a field :math:`k`. Every :math:`g\in
G` can be written in the form :math:`uh\overdot wu'` where

1. :math:`u` is a unipotent element written in the form :math:`\prod_{r=1}^N
x_r(t_r)`, with respect to a given ordering of the roots (as in Section
:ref:`SectGrpLieConstructElt`);

2. :math:`h` is a torus element represented as an element of :math:`R^d` with
each entry invertible;

3. :math:`\overdot w = \overdot s_{r_1}\cdots \overdot s_{r_k}` where
:math:`s_{r_1}\cdots s_{r_k}` is a reduced word for :math:`w` in the Weyl group.

4. :math:`u'=\prod_{r\in\Phi_w^-}x_r(t_r')` where :math:`\Phi_w^-= \{ r \mid
\hbox{$\alpha_r\in\Phi^+$ and $\alpha_rw^{-1}\in\Phi^-$}\}` and the terms are in
the usual order.

.. _SubsectGrpLieBruhat:

.. _twisted:

Twisted Groups of Lie type
--------------------------

Let :math:`G` be a connected reductive linear algebraic group defined over the
field :math:`k`. We say that :math:`H` is a *form* of :math:`G` if there is a
:math:`\bar{k}`-isomorphism between :math:`G` and :math:`H`, where
:math:`\bar{k}` the algebraic closure of :math:`k`. If some maximal torus of
:math:`G(\bar{k})` is a :math:`k`-split torus, we say that :math:`G` is *split*,
otherwise :math:`G` is *twisted*. If :math:`G` has a Borel subgroup defined over
:math:`k`, we say that :math:`G` is *quasisplit*. There is a unique split form
of every reductive linear algebraic group.

The group :math:`\Gamma := {\operatorname{Gal}}(\bar{k}:k)` acts on :math:`G` in
the usual way and :math:`G` is a :math:`\Gamma`-group in the sense of the
Section :ref:`SectGrpCohom`. The group
:math:`{\operatorname{Aut}}(G)` of algebraic automorphisms of :math:`G` is also
a :math:`\Gamma`-group. The twisted forms of :math:`G` are in one-to-one
correspondence with the :math:`1`-cocycles of :math:`\Gamma` on
:math:`{\operatorname{Aut}}(G)` and the forms are conjugate if and only if the
cocycles are cohomologous. For practical purposes it is sufficient to compute
the cohomology of :math:`\Gamma={\operatorname{Gal}}(K:k)` on
:math:`{\operatorname{Aut}}_K(G)` for some finite Galois extension :math:`K` of
:math:`k`, where :math:`{\operatorname{Aut}}_K(G)` is the group of
:math:`K`-algebraic automorphisms of :math:`G`.

The action of :math:`\Gamma` on :math:`G` induces an action on the root datum of
:math:`G`, and so we get an extended root datum. If :math:`G` is quasisplit,
then it is determined by the extended root datum and the action of
:math:`\Gamma` on :math:`K`. In general, a cocycle is required to fully
determine :math:`G`.