Operations on Elements

g * h : GrpLieElt, GrpLieElt -> GrpLieElt

The product of g and h.

Example GrpLie_GrpLieEltProduct (H86E4)

> G := GroupOfLieType( "G2", GF(3) );
> V := VectorSpace(GF(3),2);
> SetNormalising( ~G, false );
> g := elt< G | 1,2,1,2, V![2,2], <1,2>,<5,1> >;
> h := elt< G | <3,2>, V![1,2], 1 >;
> g * h;
n1 n2 n1 n2 (2 2) x1(2) x5(1) x3(2) (1 2) n1 
> SetNormalising( ~G, true ); 
> g * h;
x2(1) x3(1) (2 2) n1 n2 n1 n2 n1 x4(1) 

g ^ n : GrpLieElt, RngIntElt -> GrpLieElt

The nth power of g.

g ^ h : GrpLieElt, GrpLieELt -> GrpLieElt

The conjugate of h^(-1)gh.

( g, h ) : GrpLieElt, GrpLieELt -> GrpLieElt

The commutator g^(-1)h^(-1)gh of g and h.

Normalise( g ) : GrpLieElt ->

Normalise( g ) : GrpLieElt -> GrpLieElt

Normalise the element g. The procedural form is slightly more efficient than the functional form. If the Normalise flag is set for G, the procedure and function are redundant.

Example GrpLie_GrpLieEltArith (H86E5)

> k<z> := GF(4);
> G := GroupOfLieType( "C3", k );    
> V := VectorSpace( k, 3 );
> g := elt< G | 1,2,3, <3,z>,<4,z^2>, V![1,z^2,1] >;               
> g;
n1 n2 n3 x3(z) x4(z^2) (  1 z^2   1) 
> h := elt< G | [0,1,z,1,0,z^2,1,1,z] >;
> h;
x2(1) x3(z) x4(1) x6(z^2) x7(1) x8(1) x9(z) 
> g * h^-1;
x3(1) x5(z) x6(z^2) x8(1) (z^2 z^2   z) n1 n2 n3 x3(z^2) x5(z^2) 
> g^3;
x3(z) x5(1) x7(z^2) x8(z^2) (  1   1   z) n1 n2 n3 n1 n2 n3 n1 n2 n3 x1(1) 
x2(z^2) x3(1) x4(z) x7(z) x9(z) 

Bruhat( g ) : GrpLieElte -> GrpLieElt, GrpLieElt, GrpLieElt, GrpLieElt

The Bruhat decomposition of g. This function returns elements u, h, /dot w, u' with the properties described in Subsection Bruhat Normalisation and so that g=uh/dot wu'.

Example GrpLie_Bruhat (H86E6)

> k<z> := GF(4);
> G := GroupOfLieType( "C3", k );    
> V := VectorSpace( k, 3 );
> g := elt< G | 1,2,3, <3,z>,<4,z^2>, V![1,z^2,1] >;               
> Normalise( g ); 
x7(z^2) x8(z^2) (z^2 z^2   z) n1 n2 n3 x3(1) x6(z) 
> u, h, w, up := Bruhat( g );
> u; h; w; up;
x7(z^2) x8(z^2) 
(z^2 z^2   z) 
n1 n2 n3 
x3(1) x6(z) 

Random( G ) : GrpLie -> GrpLieElt

A uniformly random element of the group of Lie type G. The base ring of G must be finite.