Operations on Elements
Basic Operations
- g * h: GrpLieElt, GrpLieElt -> GrpLieElt
The product of two elements of a group of Lie type. If the
Normalisingflag is set for the group, then the product is normalised using the algorithms of [CHM08, CMT04]. Otherwise, the words are just concatenated.
- Example: Normalised and Unnormalised Products (ex-e7b37f)
If the
Normalisingflag 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)
- 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
Normaliseflag is set for \(G\), this operation has no effect. This uses the algorithms of [CHM08, CMT04].
- Example: Arithmetic in Groups of Lie Type (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)
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 Decomposition of an Element (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)
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.