Operations on Elements#

Basic Operations#

g * h: GrpLieElt, GrpLieElt -> GrpLieElt#: The product of two elements of a group of Lie type. If the GrpLie:Normalising flag is set for the group, then the product is normalised using the algorithms of [Cohen et al., 2008, Cohen et al., 2004]. Otherwise, the words are just concatenated.

Example: Grp Lie Elt Product (ex-e7b37f)#

If the Normalising flag is set, the product is normalised, otherwise multiplication is just concatenation.

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

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g ^ -1: GrpLieElt -> GrpLieElt#
Inverse(G): GrpLieElt -> GrpLieElt#: The inverse of the element \(g\) of a group of Lie type.

g ^ n: GrpLieElt, RngIntElt -> GrpLieElt#: The \(n\)th power of the element \(g\) of a group of Lie type.

g ^ h: GrpLieElt, GrpLieElt -> GrpLieElt#: The conjugate \(h^{-1}gh\), where \(g\) and \(h\) are elements of a group of Lie type.

(g, h): GrpLieElt, GrpLieElt -> GrpLieElt#
Commutator(g, h): GrpLieElt, GrpLieElt -> GrpLieElt#: The commutator \(g^{-1}h^{-1}gh\) of \(g\) and \(h\), where \(g\) and \(h\) are elements of a group of Lie type.

Normalise(~g): GrpLieElt#
Normalize(~g): GrpLieElt#
Normalise(g): GrpLieElt -> GrpLieElt#
Normalize(g): GrpLieElt -> GrpLieElt#: Normalise the element \(g\) of a group of Lie type \(G\). The procedural form is slightly more efficient than the functional form. If the Normalise flag is set for \(G\), this operation has no effect. This uses the algorithms of [Cohen et al., 2008, Cohen et al., 2004].

Example: Grp Lie Elt Arith (ex-6ee091)#

Arithmetic in groups of Lie type.

> 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;
x7(z^2) x8(z^2) (  z   1   z) n1 n2 n3 x3(z) x6(z^2)
> h := elt< G | [0,1,z,1,0,z^2,1,1,z] >;
> h;
x3(z) x7(1) x6(z^2) x8(1) x9(z) x2(1) x4(1)
> g * h^-1;
x3(z) x7(z^2) x6(z) x8(1) (  z   1   z) n1 n2 n3 x5(z) x6(z)
> g^3;
x3(z) x5(z) x7(z^2) x8(z^2) (z^2 z^2 z^2) n1 n2 n1 n3 n2 n1 n3 n2 n3
x3(z) x5(z^2) x7(z) x6(1) x8(z^2) x2(z) x4(1) x1(z)

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Decompositions#

Bruhat(g): GrpLieElt -> GrpLieElt, GrpLieElt, GrpLieElt, GrpLieElt#: Given an element \(g\) of a group of Lie type the Bruhat decomposition of \(g\) is returned. The function returns elements \(u\), \(h\), \(\dot{w}\), \(u'\) with the properties described in Subsection Bruhat Normalisation and so that \(g=uh\dot{w}u'\).

Example: Bruhat (ex-95f5c0)#

> 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   1   z) n1 n2 n3 x3(z) x6(z^2)
> u, h, w, up := Bruhat(g);
> u; h; w; up;
x7(z^2) x8(z^2)
(  z   1   z)
n1 n2 n3
x3(z) x6(z^2)

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MultiplicativeJordanDecomposition(x): GrpLieElt -> GrpLieElt, GrpLieElt#: The multiplicative Jordan decomposition of the element \(x\) of the group of Lie type.

Conjugacy and Cohomology#

ConjugateIntoTorus(g): GrpLieElt -> GrpLieElt, GrpLieElt#: Given a semisimple element \(g\) in a finite group of Lie type, return a torus element \(t\) and conjugator \(x\) such that \(t=xgx^{-1}\). The elements returned may be defined over a larger field than the input element.

ConjugateIntoBorel(g): GrpLieElt -> GrpLieElt, GrpLieElt#: Given a semisimple element \(g\) in a finite group of Lie type, return a Borel element \(b\) and conjugator \(x\) such that \(b=xgx^{-1}\). The elements returned may be defined over a larger field that the input element. Although any element of a group of Lie type can be conjugated into the Borel subgroup, this function is currently only implemented for semisimple elements.

Lang(c, q): GrpLieElt, RngIntElt -> GrpLieElt#: Given an element \(c\) in a finite group of Lie type and \(q\) a power of the characteristic, return a solution \(a\) of the Lang equation \(c = a^{-F} a\). Here \(F\) is the Frobenius automorphism gotten by taking \(q\)th powers in the field.