Building Groups of Lie Type

Building Groups of Lie Type#

Currently the only subgroups of a group of Lie type that can be constructed are subsystem subgroups.

SubsystemSubgroup(G, a): GrpLie, SetEnum → RootDtm#: The subsystem subgroup of the group of Lie type \(G\) generated by the standard maximal torus and the root subgroups with roots \(\alpha_{a_1},\dots,\alpha_{a_k}\) where \(a=\{a_1,\dots,a_k\}\) is a set of integers.

SubsystemSubgroup(G, s): GrpLie, SeqEnum → RootDtm#: The subsystem subgroup of the group of Lie type \(G\) generated by the standard maximal torus and the root subgroups with roots \(\alpha_{s_1},\dots,\alpha_{s_k}\) where \(s=[s_1,\dots,s_k]\) is a sequence of integers. In this version the roots must be simple in the root subdatum (i.e. none of them may be a summand of another) otherwise an error is signalled. The simple roots will appear in the subdatum in the given order.

Example: Root Subdata#

> G := GroupOfLieType("A4",Rationals());
> PositiveRoots(G);
{@
    (1 0 0 0),
    (0 1 0 0),
    (0 0 1 0),
    (0 0 0 1),
    (1 1 0 0),
    (0 1 1 0),
    (0 0 1 1),
    (1 1 1 0),
    (0 1 1 1),
    (1 1 1 1)
@}
> H := SubsystemSubgroup(G, [6,1,4]);
> H;
H: Group of Lie type A3 over Rational Field
> PositiveRoots(H);
{@
    (0 1 1 0),
    (1 0 0 0),
    (0 0 0 1),
    (1 1 1 0),
    (0 1 1 1),
    (1 1 1 1)
@}
> h := elt<H|<2,2>,1>;
> h;  G!h;
x2(2) n1
x1(2) ( 1 -1  1 -1) n2 n3 n2

DirectProduct(G1, G2): GrpLie, GrpLie → GrpLie#: The direct product of the groups \(G_1\) and \(G_2\). The two groups must have the same base ring.

Dual(G): GrpLie → GrpLie#: The dual of the group of Lie type \(G\), obtained by swapping the roots and coroots.

SolubleRadical(G): GrpLie → GrpLie#: The soluble radical of the group of Lie type \(G\).

StandardMaximalTorus(G): GrpLie → GrpLie#: The standard maximal torus of the group of Lie type \(G\).

Example: Direct Product Dual Radical#

> G1 := GroupOfLieType( "A5", GF(7) );
> G2 := GroupOfLieType( "B4", GF(7) );
> DirectProduct(G1, Dual(G2));
$: Group of Lie type A5 C4 over Finite field of size 7
>
> G := GroupOfLieType(StandardRootDatum("A",3), GF(17));
> SolubleRadical(G);
$: Torus group of Dimension 1 over Finite field of size 17

Building Groups of Lie Type

Contents

Building Groups of Lie Type#