Introduction

This chapter describes Magma functions for computing with groups of Lie type. These functions are based on [Cohen et al., 2004] for split types, and [Haller, 2005] for twisted types.

Given an extended root datum and ring with a \(\Gamma\)-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).

The Steinberg Presentation

The approach to computation in split groups of Lie type described here is based on the Steinberg presentation:cite:[Steinberg, 1962] Let \(G\) be a split group of Lie type with root datum \(R\) over the ring \(k\). Suppose the roots of \(R\) are \(\alpha_1,\dots,\alpha_{2N}\) ordered as in Section Roots, Coroots and Weights and \(n\) is the rank of \(R\). Then \(G\) contains root elements \(x_r(t)=x_{\alpha_r}(t)\) for \(t\) in \(k\). If \(R\) is semisimple, the root elements generate \(G\). In the general case, it is necessary to introduce extra torus elements. Let \(Y={\mathbb{Z}}^d\) be the coroot space of the root datum. The torus is taken to be the abelian group \(Y\otimes k^\times\), represented as the set of vectors in \(k^d\) with each component invertible, and multiplication is performed componentwise. The Weyl group of \(G\) is just the Coxeter group of the root datum \(R\). Redundant generators \(n_r\) are also included, corresponding to the generators \(s_r\) of the Weyl group.

Since the generating set is parametrised by field elements it is generally not possible to define \(G\) 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 ChapFldRe), no attempt has been made to control rounding error in this case.

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 \(k\). Every \(g\in G\) can be written in the form \(uh\overdot wu'\) where

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

2. \(h\) is a torus element represented as an element of \(R^d\) with each entry invertible;

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

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

Twisted Groups of Lie type

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

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

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