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GROUPS OF LIE TYPE

 
Acknowledgements
 
Introduction
      The Steinberg Presentation
      Bruhat Normalisation
      Twisted Groups of Lie type
 
Constructing Groups of Lie Type
      Split Groups
      Galois Cohomology
      Twisted Groups
 
Operations on Groups of Lie Type
 
Properties of Groups of Lie Type
 
Constructing Elements
 
Operations on Elements
      Basic Operations
      Decompositions
      Conjugacy and Cohomology
 
Properties of Elements
 
Roots, Coroots and Weights
      Accessing Roots and Coroots
      Reflections
      Operations and Properties for Root and Coroot Indices
      Weights
 
Building Groups of Lie Type
 
Automorphisms
      Basic Functionality
      Constructing Special Automorphisms
      Operations and Properties of Automorphisms
 
Algebraic Homomorphisms
 
Twisted Tori
 
Sylow Subgroups
 
Representations
 
Bibliography







DETAILS

 
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 (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)

 
Operations on Elements

      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

 
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 (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)

 
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 (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