|
The following functions construct the standard automorphisms of a group of Lie
type, as described in [Car72] (except for the graph
automorphism of G2). In many cases, including the finite
groups, every automorphism is a product of these standard automorphisms.
Automorphism group of a group of Lie type G.
One(A) : GrpLieAuto -> GrpLieAutoElt
Id(A) : GrpLieAuto -> GrpLieAutoElt
The identity automorphism of the group of Lie type G.
The map object associated with the automorphism a.
Given a map object m from G to G, which is an isomorphism,
returns the associated automorphism as an automorphism of a group
of Lie type.
The composition of the group of Lie type automorphisms h and g.
The nth power of the group of Lie type automorphism h.
The conjugate h - 1gh, where g and h are group of Lie type automorphisms g and h
Codomain(A) : GrpLieAuto -> GrpLie
Domain(h) : GrpLieAutoElt -> GrpLie
Codomain(h) : GrpLieAutoElt -> GrpLie
Domain or codomain of an automorphism of a group of
Lie type or of the group of automorphisms.
The inner automorphism taking g∈G to gx, where x is an element of the
group of Lie type G.
The diagonal automorphism of the semisimple group of Lie type G given by the
vector v.
Let n be the semisimple rank of G and let k be its base field.
Then v must be a vector in kn with every component nonzero.
The function returns the automorphism given by the character χ defined by
χ(αi)=vi, where αi is the ith simple root.
Since our groups are algebraic, a diagonal automorphism is just a special
case of an inner automorphism.
DiagramAutomorphism(G, p) : GrpLie, GrpPermElt -> Map
SimpleSigns: Any Default: 1
The graph automorphism of the group of Lie type G given by the permutation p.
The permutation must act on the indices of simple roots of G or the indices of
all roots of G. The graph automorphism of the group of type G2 has not
been implemented yet.
The optional parameter SimpleSigns can be used to specify the signs
corresponding to each simple root. This should either be a sequence
of integers ∓ 1, or a single integer ∓ 1.
The field automorphism of the group of Lie type G induced by σ,
an element of the automorphism group of the base field of G
Random(A) : GrpLieAuto -> GrpLieAutoElt
A random element in A, the automorphism group
of the group of Lie type G.
The duality automorphism of G.
This is an automorphism that takes every unipotent term xr(t) to
xs(∓ t), where s= Negative(RootDatum(G),r)).
The Frobenius automorphism of the finite group of Lie type G gotten by qth
powers in the base field.
The integer q must be a power of the characteristic of the base field of G.
Given a group of Lie type automorphism h, this returns
a field automorphism f,
a graph automorphism g and
an inner automorphism i
such that h=fgi. This only works for groups defined over finite fields.
The algorithm is due to Scott Murray and Sergei Haller.
Returns true if and only if the automorphism h is algebraic.
Some automorphisms of B 2(4)
> G := GroupOfLieType("B2", GF(4));
> A := AutomorphismGroup(G);
> A!1 eq IdentityAutomorphism(G);
true
> g := GraphAutomorphism(G, Sym(2)!(1,2));
> g;
Automorphism of Group of Lie type B2 over Finite field of size 2^2
given by: Mapping from: Group of Lie type to Group of Lie type
given by a rule
Decomposition:
Mapping from: GF(2^2) to GF(2^2) given by a rule,
(1, 2),
1
The automorphism of B 2(4) whose stabiliser is () 2B 2(4) is constructed
by the following code.
> sigma := iso< GF(4) -> GF(4) | x :-> x^2, x :-> x^2 >;
> h := FieldAutomorphism(G, sigma) * g;
> h in A;
true
> f,g,i := DecomposeAutomorphism(h);
> assert f*g*i eq h;
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|