- Introduction
- 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 (H110E1)
- Galois Cohomology
- Twisted Groups
- TwistedGroupOfLieType(c) : OneCoC -> GrpLie
- TwistedGroupOfLieType(R, k, K) : RootDtm, Rng, Rng-> GrpLie
- TwistedGroupOfLieType(R, q, r) : RootDtm, RngIntElt, RngIntElt -> GrpLie
- TwistedGroupOfLieType(t, r, q) : MonStgElt, RngIntElt, RngIntElt -> GrpLie
- Example GrpLie_TwistedGrpLieType1 (H110E3)
- BaseRing(G) : GrpLie -> Rng
- DefRing(G) : GrpLie -> Rng
- UntwistedOvergroup(G) : GrpLie -> GrpLie
- Example GrpLie_TwistedGrpLieType2 (H110E4)
- RelativeRootElement(G,delta,t) : GrpLie, RngIntElt, [FldElt] -> GrpLieElt
- Example GrpLie_RelativeRootElts (H110E5)
- 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 (H110E6)
- Order(G) : GrpLie -> RngIntElt
- FactoredOrder(G) : GrpLie -> RngIntElt
- Dimension(G) : GrpLie -> RngIntElt
- Example GrpLie_Orders (H110E7)
- 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
- Constructing Elements
- Operations on Elements
- Properties of Elements
- Roots, Coroots and Weights
- Building Groups of Lie Type
- Automorphisms
- Basic Functionality
- 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
- Algebraic Homomorphisms
- Twisted Tori
- Sylow Subgroups
- Representations
- Curtis--Steinberg--Tits Presentations
- CST_Generators(t,r,q,w) : MonStgElt, RngIntElt, RngIntElt, SeqEnum -> SeqEnum, SeqEnum
- CST_Presentation(t,r,q) : MonStgElt, RngIntElt, RngIntElt -> GrpSLP, SeqEnum
- CST_VerifyPresentation(t,r,q,X,Y) : MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum -> BoolElt, RngIntElt
- Example GrpLie_CSTPres (H110E25)
- CSTtoChev(t,r,q,X,Y) : MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum -> Map
- Example GrpLie_CSTtoChev (H110E26)
- ExtendGeneratorList(t,r,X,Y) : MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum -> SeqEnum,SeqEnum
- IrreducibleHighestWeightRepresentation(G,w) : GrpLie, SeqEnum -> Map
- IrreducibleHighestWeightGenerators(G,w) : GrpLie, SeqEnum -> SeqEnum,SeqEnum
- IrreducibleHighestWeightFunction(G,w) : GrpLie, SeqEnum -> UserProgram
- VermaModule(G,w) : GrpLie, SeqEnum -> ModGrp
- UniversalHighWeightRepresentation(G,w) : GrpLie, SeqEnum -> Map,SeqEnum,SeqEnum
- Chevalley Groups
- StandardLieRepresentation(t,r) : MonStgElt, RngIntElt -> SeqEnum, SeqEnum
- AdjointChevalleyGroup(t,r,q) : MonStgElt,RngIntElt,RngIntElt -> GrpMat
- Example GrpLie_AdjointChev (H110E27)
- LieRootMatrix(R,α,B) : RootDtm,ModTupFldElt,SetIndx -> AlgMatElt
- LieTypeGenerators(t,k,q) : MonStgElt, RngIntElt, RngIntElt -> SeqEnum,SeqEnum
- SLPGeneratorList(t,r,q) : MonStgElt, RngIntElt, RngIntElt -> SeqEnum, SeqEnum
- Morphisms and the Row Reduction Algorithm
- Morphism(G,X,Y) : GrpLie,SeqEnum,SeqEnum -> Map
- ChevalleyForm(ρ,A) : Map[GrpLie,GrpMat], GrpMatElt -> SeqEnum, FldFinElt
- Example GrpLie_ChevForm (H110E28)
- PrepareRewrite(t,r,q,X,Y) : MonStgElt,RngIntElt,RngIntElt,SeqEnum,SeqEnum -> UserProgram, Map
- LieTypeRewrite(t,r,q,X,Y,g) : MonStgElt,RngIntElt,RngIntElt,SeqEnum,SeqEnum,GrpMatElt -> BoolElt, GrpSLPElt
- Example GrpLie_LieRewrite (H110E29)
- RowReductionMap(ρ) : Map[GrpLie,GrpMat] -> UserProgram
- Bibliography
V2.28, 13 July 2023