Constructing Groups of Lie type
GroupOfLieType( n, R ) : MonStgElt, Rng -> AlgMatElt
GroupOfLieType( W, R ) : GrpCox, Rng -> AlgMatElt
GroupOfLieType( RD, R ) : RootDtm, Rng -> AlgMatElt
GroupOfLieType( C, R ) : AlgMatElt, Rng -> AlgMatElt
IsNormalising( G ) : GrpLie -> BoolElt
SetNormalising( G, Normalising ) : GrpLie, BoolElt -> .
Example GrpLie_Create (H35E1)
Constructing Elements
elt<G | L> : GrpLie, List -> GrpMatElt
TorusTerm( G, r, t ) : GrpLie, RngIntElt, . -> GrpLieElt
Eltlist( g ) : GrpLieElt -> List
Example GrpLie_ElementCreate (H35E2)
Operations on Groups of Lie type
G eq H : GrpLie, GrpLie -> BoolElt
IsAlgebraicallyIsomorphic( G, H ) : GrpLie, GrpLie -> BoolElt
IsIsogenous( G, H ) : GrpLie, GrpLie -> BoolElt
Generators( G ) : GrpLie ->
AlgebraicGenerators( G ) : GrpLie ->
Example GrpLie_Generators (H35E3)
WeylGroup( G ) : GrpLie -> GrpCox
BaseRing( G ) : GrpLie -> Rng
RootDatum( G ) : GrpLie -> RootDtm
Rank( G ) : GrpLie -> RngIntElt
SemisimpleRank( G ) : GrpLie -> RngIntElt
CartanMatrix( G ) : GrpLie -> AlgMatElt
CartanName( G ) : GrpLie -> MonStgElt
DynkinDiagram( G ) : GrpLie -> .
FundamentalGroup( G ) : GrpLie -> RootDtm
IsogenyGroup( G ) : GrpLie -> RootDtm
CoisogenyGroup( G ) : GrpLie -> RootDtm
NumberOfPositiveRoots( G ) : GrpLie -> RngIntElt
Roots( G ) : GrpLie -> {@@}
PositiveRoots( G ) : GrpLie -> {@@}
Root( G, r ) : GrpLie, RngIntElt -> {@@}
RootPosition( G, v ) : GrpLie, . -> {@@}
CoxeterElement( G ) : GrpCox -> GrpPermElt
CoxeterNumber( G ) : GrpCox -> GrpPermElt
WeightLattice( G ) : RootDtm -> Lat
FundamentalWeights( G ) : GrpLie -> SeqEnum
Properties of Groups of Lie type
IsSimple( G ) : GrpLie -> BoolElt
IsSimplyLaced( G ) : GrpLie-> BoolElt
IsSemisimple( G ) : GrpLie-> BoolElt
IsAdjoint( G ) : GrpLie-> BoolElt
IsSimplyConnected( G ) : GrpLie-> BoolElt
Operations on Elements
g * h : GrpLieElt, GrpLieElt -> GrpLieElt
Example GrpLie_GrpLieEltProduct (H35E4)
Inverse( g ) : GrpLieElt -> GrpLieElt
Identity( G ) : GrpLie -> GrpLieElt
g ^ n : GrpLieElt, RngIntElt -> GrpLieElt
g ^ h : GrpLieElt, GrpLieELt -> GrpLieElt
( g, h ) : GrpLieElt, GrpLieELt -> GrpLieElt
Normalise( g ) : GrpLieElt ->
Example GrpLie_GrpLieEltArith (H35E5)
Bruhat( g ) : GrpLieElte -> GrpLieElt, GrpLieElt, GrpLieElt, GrpLieElt
Example GrpLie_Bruhat (H35E6)
Random( G ) : GrpLie -> GrpLieElt
Matrix Representations
StandardRepresentation( G ) : GrpLie -> Map
Example GrpLie_StandardRepresentation (H35E7)
[Future release] RegularRepresentation( G ) : GrpLie -> Map