Constructing Elements

elt<G | L> : GrpLie, List -> GrpMatElt
Given a group of Lie type G over the ring R and a list L of appropriate objects, construct an element of G. Suppose the underlying root datum has dimension d, rank n, and roots α1, ..., α2N. Each entry in the list can be one of the following:
1.
A tuple <r, t> where r=1, ..., 2N and t∈R. This corresponds to the unipotent term xr(t).
2.
A sequence of tuples as in item (1).
3.
A sequence [t1, ..., tN] of elements of R. This corresponds to the unipotent element x1(t1) ... xN(tN).
4.
An integer r=1, ..., 2N. This corresponds to the Weyl group representative nr.
5.
A Weyl group element w, either as a word or as a permutation. This corresponds to the Weyl group representative /dot w.
6.
A vector v∈Rd with each entry invertible. This corresponds to an element of the torus.
7.
An element of G, either previously constructed or (recursively) given as a list of terms of the forms 1 to 7.
Identity(G) : GrpLie -> GrpLieElt
Id(G) : GrpLie -> GrpLieElt
G ! 1 : GrpLie, RngIntElt -> GrpLieElt
elt< G | > : GrpLie -> GrpLieElt
The identity element of the group of Lie type G.

Example GrpLie_ElementCreate (H110E8)

> G := GroupOfLieType("A5", Rationals() : Normalising := false);
> V := VectorSpace(Rationals(), 5);
> NumPosRoots(G);
15
> elt< G | <5,1/2>, 1,3,2, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],
>          V![6,1/3,-1,3,2/3] >;
x5(1/2) n1 n3 n2 x1(1) x2(2) x3(3) x4(4) x5(5) x6(6) x7(7) x8(8) x9(9)
x10(10) x11(11) x12(12) x13(13) x14(14) x15(15) ( 6 1/3  -1   3 2/3)
TorusTerm(G, r, t) : GrpLie, RngIntElt, RngElt -> GrpLieElt
The torus term hr(t)=αrstar tensor t in the group of Lie type G, where r is the index of the coroot αrstar and t an element of the base ring of G.
CoxeterElement(G) : GrpLie -> GrpPermElt
The Coxeter elementof the group of Lie type G, i.e. the representative of the Coxeter element in the Weyl group of G.
Random(G) : GrpLie -> GrpLieElt
    Uniform: BoolElt                    Default: true
An element of the (twisted or untwisted) finite group of Lie type G chosen at random. The base ring of G must be finite. If the optional parameter Uniform is set to true, the random elements to be distributed uniformly. If the optional parameter Uniform is set to false, this function is much faster but the random elements are not distributed uniformly. Instead each double coset of the Borel subgroup occurs with equal frequency, and the elements are uniformly distributed within each double coset.
Eltlist(g) : GrpLieElt -> List
The list corresponding to the element g of a group of Lie type.
CentrePolynomials(G) : GrpLie ->
CenterPolynomials(G) : GrpLie ->
A set of polynomials which are satisfied by the coordinates of a torus element h of the group of Lie type G if, and only if, h is in the centre of G.

Example GrpLie_Centre (H110E9)

The centre of a semisimple group is finite, so the centre polynomials can be used to find all central elements. 
> G := GroupOfLieType("B3", Rationals() : Isogeny:="SC");
> pols := CentrePolynomials(G);
> pols;
{
    -h[2] + h[3]^2,
    h[1]^2 - h[2],
    -h[1]*h[3]^2 + h[2]^2
}
> S := Scheme(AffineSpace(Rationals(), 3), Setseq(pols));
> pnts := RationalPoints(S);
> pnts;
{@ (0, 0, 0), (1, 1, -1), (1, 1, 1) @}
The rational points of S can be converted into elements of G, taking care to eliminate any point which has a coordinate equal to zero: 
> V := VectorSpace(Rationals(), 3);
> [ elt< G | V!Eltseq(pnt) > : pnt in pnts | &*Eltseq(pnt) ne 0 ];
[ (1  1 -1) , 1 ]
V2.28, 13 July 2023