Introduction

      The Steinberg Presentation

      Bruhat Normalisation

      Twisted Groups of Lie type

 
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 (H113E1)

      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 (H113E2)

      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 (H113E3)
            BaseRing(G) : GrpLie -> Rng
            DefRing(G) : GrpLie -> Rng
            UntwistedOvergroup(G) : GrpLie -> GrpLie
            Example GrpLie_TwistedGrpLieType2 (H113E4)
            RelativeRootElement(G,delta,t) : GrpLie, RngIntElt, [FldElt] -> GrpLieElt
            Example GrpLie_RelativeRootElts (H113E5)

 
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 (H113E6)
      Order(G) : GrpLie -> RngIntElt
      FactoredOrder(G) : GrpLie -> RngIntElt
      Dimension(G) : GrpLie -> RngIntElt
      Example GrpLie_Orders (H113E7)
      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 (H113E8)
      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 (H113E9)

 
Operations on Elements

      Basic Operations
            g * h : GrpLieElt, GrpLieElt -> GrpLieElt
            Example GrpLie_GrpLieEltProduct (H113E10)
            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 (H113E11)

      Decompositions
            Bruhat(g) : GrpLieElt -> GrpLieElt, GrpLieElt, GrpLieElt, GrpLieElt
            Example GrpLie_Bruhat (H113E12)
            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

 
Roots, Coroots and Weights

      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 (H113E13)
            HighestRoot(G) : GrpLie -> LatElt
            HighestShortRoot(G) : GrpLie -> LatElt
            Example GrpLie_HeighestRoots (H113E14)

      Reflections
            Reflections(G) : GrpLie -> GrpLieElt
            Reflection(G, r) : GrpLie, RngIntElt -> GrpLieElt
            Example GrpLie_Reflections (H113E15)

      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 (H113E16)

      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 (H113E17)
      DirectProduct(G1, G2) : GrpLie, GrpLie -> GrpLie
      Dual(G) : GrpLie -> GrpLie
      SolubleRadical(G) : GrpLie -> GrpLie
      StandardMaximalTorus(G) : GrpLie -> GrpLie
      Example GrpLie_DirectProductDualRadical (H113E18)

 
Automorphisms

      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 (H113E19)

 
Algebraic Homomorphisms
      GroupOfLieTypeHomomorphism(phi, k) : Map, Rng -> .
      Example GrpLie_CreatingRootDataHomomorphisms (H113E20)

 
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 (H113E21)
      Example GrpLie_GrpLieTori2 (H113E22)

 
Sylow Subgroups
      PrintSylowSubgroupStructure(G) : GrpLie ->
      SylowSubgroup(G, p) : GrpLie, RngIntElt -> List
      Example GrpLie_GrpLieSylow (H113E23)

 
Representations
      StandardRepresentation(G) : GrpLie -> Map
      AdjointRepresentation(G) : GrpLie -> Map, AlgLie
      LieAlgebra(G) : GrpLie -> AlgLie, Map
      HighestWeightRepresentation(G, v) : GrpLie, . -> Map
      Example GrpLie_StandardRepresentation (H113E24)
      ContravariantForm(ρ) : Map[GrpLie,GrpMat] -> AlgMatElt
      GeneralisedRowReduction(ρ) : Map -> Map

 
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 (H113E25)
      CSTtoChev(t,r,q,X,Y) : MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum -> Map
      Example GrpLie_CSTtoChev (H113E26)
      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 (H113E27)
            LieRootMatrix(R,α,B) : RootDtm,ModTupFldElt,SetIndx -> AlgMatElt
            LieRootMatrix(R,r,X) : RootDtm, RngIntElt, SeqEnum -> 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 (H113E28)
            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 (H113E29)
            RowReductionMap(ρ) : Map[GrpLie,GrpMat] -> UserProgram

 
Bibliography

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