Constructing Groups of Lie Type
Split Groups
GroupOfLieType(N, k) : MonStgElt, Rng -> GrpLie
GroupOfLieType(N, q) : MonStgElt, RngIntElt -> GrpLie
GroupOfLieType(W, k) : GrpPermCox, Rng -> GrpLie
GroupOfLieType(W, q) : GrpPermCox, RngIntElt -> GrpLie
GroupOfLieType(R, k) : RootDtm, Rng -> GrpLie
GroupOfLieType(R, q) : RootDtm, RngIntElt -> GrpLie
GroupOfLieType(C, k) : Mtrx, Rng -> GrpLie
GroupOfLieType(C, q) : Mtrx, RngIntElt -> GrpLie
SimpleGroupOfLieType(X, n, k) : MonStgElt, RngIntElt, Rng -> GrpLie
SimpleGroupOfLieType(X, n, q) : MonStgElt, RngIntElt, RngIntElt -> GrpLie
GroupOfLieType(L) : AlgLie -> GrpLie
IsNormalising(G) : GrpLie -> BoolElt
Example GrpLie_Create (H103E1)
Galois Cohomology
GammaGroup(k, G) : Fld, GrpLie -> GGrp
GammaGroup(k, A) : Fld, GrpLieAuto -> GGrp
ActingGroup(G) : GrpLie -> Grp, Map
ExtendGaloisCocycle(c) : OneCoC -> OneCoC
GaloisCohomology(A) : GGrp -> SeqEnum
IsInTwistedForm(x, c) : GrpLieElt, OneCoC -> BoolElt
Example GrpLie_GalCohom (H103E2)
Twisted Groups
TwistedGroupOfLieType(c) : OneCoC -> GrpLie
TwistedGroupOfLieType(R, k, K) : RootDtm, Rng, Rng-> GrpLie
BaseRing(G) : GrpLie -> Rng
DefRing(G) : GrpLie -> Rng
UntwistedOvergroup(G) : GrpLie -> GrpLie
Example GrpLie_TwistedGrpLieType (H103E3)
RelativeRootElement(G,delta,t) : GrpLie, RngIntElt, [FldElt] -> GrpLieElt
Example GrpLie_RelativeRootElts (H103E4)
Operations on Groups of Lie Type
G eq H : GrpLie, GrpLie -> BoolElt
G subset H : GrpLie, GrpLie -> BoolElt
IsAlgebraicallyIsomorphic(G, H) : GrpLie, GrpLie -> BoolElt, Map
IsIsogenous(G, H) : GrpLie, GrpLie -> BoolElt
IsCartanEquivalent(G, H) : GrpLie, GrpLie -> BoolElt
BaseRing(G) : GrpLie -> Rng
BaseExtend(G, K) : GrpLie, Rng -> GrpLie, Map
ChangeRing(G, K) : GrpLie, Rng -> GrpLie
Generators(G) : GrpLie ->
NumberOfGenerators(G) : GrpLie -> RngIntElt
AlgebraicGenerators(G) : GrpLie ->
NumberOfAlgebraicGenerators(G) : GrpLie -> RngIntElt
Example GrpLie_Generators (H103E5)
Order(G) : GrpLie -> RngIntElt
FactoredOrder(G) : GrpLie -> RngIntElt
Dimension(G) : GrpLie -> RngIntElt
Example GrpLie_Orders (H103E6)
CartanName(G) : GrpLie -> Mtrx
RootDatum(G) : GrpLie -> RootDtm
DynkinDiagram(G) : GrpLie ->
CoxeterDiagram(G) : GrpLie ->
CoxeterMatrix(G) : GrpLie -> AlgMatElt
CoxeterGraph(G) : GrpLie -> GrphUnd
CartanMatrix(G) : GrpLie -> GrphUnd
DynkinDigraph(G) : GrpLie -> GrphUnd
Rank(G) : GrpLie -> RngIntElt
SemisimpleRank(G) : GrpLie -> RngIntElt
CoxeterNumber(G) : GrpLie -> RngIntElt
WeylGroup(G) : GrpLie -> GrpPermCox
WeylGroup(GrpFPCox, G) : Cat, GrpLie -> GrpFPCox
WeylGroup(GrpMat, G) : Cat, GrpLie -> GrpMat
FundamentalGroup(G) : GrpLie -> GrpAb, Map
IsogenyGroup(G) : GrpLie -> GrpAb, Map
CoisogenyGroup(G) : GrpLie -> GrpAb, Map
Properties of Groups of Lie Type
IsFinite(G) : GrpLie -> BoolElt
IsAbelian(G) : GrpLie -> BoolElt
IsSimple(G) : GrpLie -> BoolElt
IsSimplyLaced(G) : GrpLie-> BoolElt
IsSemisimple(G) : GrpLie-> BoolElt
IsAdjoint(G) : GrpLie -> BoolElt
IsWeaklyAdjoint(G) : GrpLie -> BoolElt
IsSimplyConnected(G) : GrpLie -> BoolElt
IsWeaklySimplyConnected(G) : GrpLie -> BoolElt
IsSplit(G) : GrpLie -> BoolElt
IsTwisted(G) : GrpLie -> BoolElt
Constructing Elements
elt<G | L> : GrpLie, List -> GrpMatElt
Identity(G) : GrpLie -> GrpLieElt
Example GrpLie_ElementCreate (H103E7)
TorusTerm(G, r, t) : GrpLie, RngIntElt, RngElt -> GrpLieElt
CoxeterElement(G) : GrpLie -> GrpPermElt
Random(G) : GrpLie -> GrpLieElt
Eltlist(g) : GrpLieElt -> List
CentrePolynomials(G) : GrpLie ->
Example GrpLie_Centre (H103E8)
Basic Operations
g * h : GrpLieElt, GrpLieElt -> GrpLieElt
Example GrpLie_GrpLieEltProduct (H103E9)
g ^ -1 : GrpLieElt -> GrpLieElt
g ^ n : GrpLieElt, RngIntElt -> GrpLieElt
g ^ h : GrpLieElt, GrpLieElt -> GrpLieElt
(g, h) : GrpLieElt, GrpLieElt -> GrpLieElt
Normalise(simg) : GrpLieElt ->
Example GrpLie_GrpLieEltArith (H103E10)
Decompositions
Bruhat(g) : GrpLieElt -> GrpLieElt, GrpLieElt, GrpLieElt, GrpLieElt
Example GrpLie_Bruhat (H103E11)
MultiplicativeJordanDecomposition(x) : GrpLieElt -> GrpLieElt, GrpLieElt
Conjugacy and Cohomology
ConjugateIntoTorus(g) : GrpLieElt -> GrpLieElt, GrpLieElt
ConjugateIntoBorel(g) : GrpLieElt -> GrpLieElt, GrpLieElt
Lang(c, q) : GrpLieElt, RngIntElt -> GrpLieElt
Properties of Elements
IsSemisimple(x) : GrpLieElt -> BoolElt
IsUnipotent(x) : GrpLieElt -> BoolElt
IsCentral(x) : GrpLieElt -> BoolElt
Accessing Roots and Coroots
RootSpace(G) : GrpLie -> Lat
SimpleRoots(G) : GrpLie -> Mtrx
NumberOfPositiveRoots(G) : GrpLie -> RngIntElt
Roots(G) : GrpLie -> (@@)
PositiveRoots(G) : GrpLie -> (@@)
Root(G, r) : GrpLie, RngIntElt -> (@@)
RootPosition(G, v) : GrpLie, . -> (@@)
Example GrpLie_RootsCoroots (H103E12)
HighestRoot(G) : GrpLie -> LatElt
HighestShortRoot(G) : GrpLie -> LatElt
Example GrpLie_HeighestRoots (H103E13)
Reflections
Reflections(G) : GrpLie -> GrpLieElt
Reflection(G, r) : GrpLie, RngIntElt -> GrpLieElt
Example GrpLie_Reflections (H103E14)
Operations and Properties for Root and Coroot Indices
RootHeight(G, r) : GrpLie, RngIntElt -> RngIntElt
RootNorms(G) : GrpLie -> [RngIntElt]
RootNorm(G, r) : GrpLie, RngIntElt -> RngIntElt
IsLongRoot(G, r) : GrpLie, RngIntElt -> BoolElt
IsShortRoot(G, r) : GrpLie, RngIntElt -> BoolElt
AdditiveOrder(G) : GrpLie -> SeqEnum
Example GrpLie_AdditiveOrder (H103E15)
Weights
WeightLattice(G) : GrpLie -> Lat
FundamentalWeights(G) : GrpLie -> Mtrx
DominantWeight(G, v) : GrpLie, . -> ModTupFldElt, GrpFPCoxElt
Building Groups of Lie Type
SubsystemSubgroup(G, a) : GrpLie, SetEnum -> RootDtm
SubsystemSubgroup(G, s) : GrpLie, SeqEnum -> RootDtm
Example GrpLie_RootSubdata (H103E16)
DirectProduct(G1, G2) : GrpLie, GrpLie -> GrpLie
Dual(G) : GrpLie -> GrpLie
SolubleRadical(G) : GrpLie -> GrpLie
StandardMaximalTorus(G) : GrpLie -> GrpLie
Example GrpLie_DirectProductDualRadical (H103E17)
Basic Functionality
AutomorphismGroup(G) : GrpLie -> GrpLieAuto
IdentityAutomorphism(G) : GrpLie -> GrpLieAutoElt
Mapping(a) : GrpLieAutoElt -> Map
Automorphism(m) : Map -> GrpLieAutoElt
h * g : GrpLieAutoElt, GrpLieAutoElt -> GrpLieAutoElt
h ^ n : GrpLieAutoElt, RngIntElt -> GrpLieAutoElt
g ^ h : GrpLieAutoElt, GrpLieAutoElt -> GrpLieAutoElt
Domain(A) : GrpLieAuto -> GrpLie
Constructing Special 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
RandomAutomorphism(G) : GrpLie -> GrpLieAutoElt
DualityAutomorphism(G) : GrpLie -> GrpLieAutoElt
FrobeniusMap(G,q) : GrpLie, RngIntElt -> GrpLieAutoElt
Operations and Properties of Automorphisms
DecomposeAutomorphism(h) : GrpLieAutoElt -> GrpLieAutoElt, GrpLieAutoElt,GrpLieAutoElt, Rec
IsAlgebraic(h) : GrpLieAutoElt -> BoolElt
Example GrpLie_Automorphism (H103E18)
Algebraic Homomorphisms
GroupOfLieTypeHomomorphism(phi, k) : Map, Rng -> .
Example GrpLie_CreatingRootDataHomomorphisms (H103E19)
Twisted Tori
TwistedTorusOrder(R, w) : RootDtm, GrpPermElt -> SeqEnum
TwistedToriOrders(G) : GrpLie -> SeqEnum
TwistedTorus(G, w) : GrpLie, GrpPermElt -> List
TwistedTori(G) : GrpLie -> SeqEnum
Example GrpLie_GrpLieTori (H103E20)
Example GrpLie_GrpLieTori2 (H103E21)
Sylow Subgroups
PrintSylowSubgroupStructure(G) : GrpLie ->
SylowSubgroup(G, p) : GrpLie, RngIntElt -> List
Example GrpLie_GrpLieSylow (H103E22)
Representations
StandardRepresentation(G) : GrpLie -> Map
AdjointRepresentation(G) : GrpLie -> Map, AlgLie
LieAlgebra(G) : GrpLie -> AlgLie, Map
HighestWeightRepresentation(G, v) : GrpLie, . -> Map
Example GrpLie_StandardRepresentation (H103E23)
GeneralisedRowReduction(ρ) : Map -> Map
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012