Constructing Groups of Lie Type
GroupOfLieType( N, k ) : MonStgElt, Rng -> AlgMatElt
GroupOfLieType( W, k ) : GrpFPCox, Rng -> AlgMatElt
GroupOfLieType( R, k ) : RootDtm, Rng -> AlgMatElt
GroupOfLieType( C, k ) : Mtrx, Rng -> AlgMatElt
IsNormalising( G ) : GrpLie -> BoolElt
SetNormalising( G, Normalising ) : GrpLie, BoolElt -> .
Example GrpLie_Create (H86E1)
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 (H86E2)
CartanName( G ) : GrpLie -> Mtrx
DynkinDiagram( G ) : GrpLie -> Mtrx
CoxeterDiagram( G ) : GrpLie -> Mtrx
WeylGroup( G ) : GrpLie -> GrpCox
CoxeterMatrix( G ) : GrpLie -> AlgMatElt
CoxeterGraph( G ) : GrpLie -> GrphUnd
CartanMatrix( G ) : GrpLie -> AlgMatElt
DynkinDigraph( G ) : GrpLie -> GrphDir
BaseRing( G ) : GrpLie -> Rng
RootDatum( G ) : GrpLie -> RootDtm
Rank( G ) : GrpLie -> RngIntElt
SemisimpleRank( G ) : GrpLie -> RngIntElt
Order( G ) : GrpLie -> RngIntElt
CartanMatrix( G ) : GrpLie -> Mtrx
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
Constructing Elements
elt<G | L> : GrpLie, List -> GrpMatElt
Identity( G ) : GrpLie -> GrpLieElt
TorusTerm( G, r, t ) : GrpLie, RngIntElt, . -> GrpLieElt
Eltlist( g ) : GrpLieElt -> List
Example GrpLie_ElementCreate (H86E3)
Operations on Elements
g * h : GrpLieElt, GrpLieElt -> GrpLieElt
Example GrpLie_GrpLieEltProduct (H86E4)
g ^ n : GrpLieElt, RngIntElt -> GrpLieElt
g ^ h : GrpLieElt, GrpLieELt -> GrpLieElt
( g, h ) : GrpLieElt, GrpLieELt -> GrpLieElt
Normalise( g ) : GrpLieElt ->
Example GrpLie_GrpLieEltArith (H86E5)
Bruhat( g ) : GrpLieElte -> GrpLieElt, GrpLieElt, GrpLieElt, GrpLieElt
Example GrpLie_Bruhat (H86E6)
Random( G ) : GrpLie -> GrpLieElt
Automorphisms
InnerAutomorphism( G, x ) : GrpLie, GrpLieElt -> Map
DiagonalAutomorphism( G, v ) : GrpLie, ModTupRngElt -> Map
GraphAutomorphism( G, p ) : GrpLie, GrpPermElt -> Map
FieldAutomorphism( G, sigma ) : GrpLie, Map -> Map
Example GrpLie_Automorphism (H86E7)
Constructing Representations
StandardRepresentation( G ) : GrpLie -> Map
Example GrpLie_StandardRepresentation (H86E8)
AdjointRepresentation( G ) : GrpLie -> Map
HighestWeightRepresentation( G, v ) : GrpLie, . -> Map
Operations on Representations
Weight( rho, v ) : Map, ModTupRngElt -> LatElt
HighestWeightVectors( rho ) : Map -> [ModTupRngElt]
HighestWeights( rho ) : Map -> [LatElt], [ModTupRngElt]
WeightVectors( rho ) : Map -> [ModTupRngElt]
Weights( rho ) : Map -> [LatElt], [ModTupRngElt]