|
[____]
This chapter describes Magma functions for computing
with groups of Lie type.
These functions are based on [CMT04] for split types, and
[Hal05] for twisted types.
Given an extended root datum and ring with a Γ-action, a group of Lie
type can be constructed in Magma. Such groups include reductive Lie
groups (when the ring is R or C), 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 approach to computation in split groups of Lie type described here is based on the
Steinberg presentation[Ste62]
Let G be a split group of Lie type with root datum R over the ring k.
Suppose
the roots of R are α1, ..., α2N ordered as in
Section Roots, Coroots and Weights and n is the rank of R. Then G contains
root elements xr(t)=xαr(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=Zd be the coroot space of the root datum. The torus is taken to be the
abelian group Y tensor k x , represented as the set of vectors in kd
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 nr are also included, corresponding to the
generators sr 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 REAL AND COMPLEX FIELDS), no
attempt has been made to control rounding error in this case.
The Bruhat decomposition [Car93, Chapter
2] gives us a useful normal form for elements of a split
group of Lie type
defined over a field k. Every g∈G can be written in the form uh/dot wu' where
- 1.
- u is a unipotent element written in the form ∏r=1N
xr(tr), with respect to a given ordering of the roots (as in
Section Constructing Elements);
- 2.
- h is a torus element represented as an element of Rd
with each entry invertible;
- 3.
- /dot w = /dot sr1 ... /dot srk where
sr1 ... srk is a reduced word for w in the Weyl group.
- 4.
- u'=∏r∈Φw^ - xr(tr') where Φw^ - = { r | hbox(α_r∈Φ^+ and α_rw^{-1}∈Φ^-}} and the terms
are in the usual order.
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 Γ := Gal(bar(k):k) acts on G in the usual way and G
is a Γ-group in the sense of the Section Finite Group Cohomology.
The group Aut(G) of algebraic automorphisms of G is also a
Γ-group.
The twisted forms of G are in one-to-one correspondence with the
1-cocycles of Γ on 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 Γ=Gal(K:k) on AutK(G) for some finite Galois extension K
of k, where AutK(G) is the group of K-algebraic automorphisms of G.
The action of Γ 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 Γ on K.
In general, a cocycle is required to fully determine G.
[Next][Prev] [Right] [____] [Up] [Index] [Root]
|