Introduction

      Reflections

      Definition of a Split Root Datum

      Simple and Positive Roots

      The Coxeter Group

      Nonreduced Root Data

      Isogeny of Split Reduced Root Data

      Extended Root Data

 
Constructing Root Data
      RootDatum(N) : MonStgElt -> RootDtm
      Example RootDtm_CreatingRootData (H107E1)
      Example RootDtm_CreatingExtendedRootData (H107E2)
      RootDatum(C) : AlgMatElt -> RootDtm
      RootDatum(D) : GrphDir -> RootDtm
      RootDatum(A, B) : Mtrx, Mtrx -> RootDtm
      Example RootDtm_G2RootSystem (H107E3)
      IrreducibleRootDatum(X, n) : MonStgElt, RngIntElt -> RootDtm
      StandardRootDatum(X, n) : MonStgElt, RngIntElt -> RootDtm
      Example RootDtm_IrreducibleRootDatum (H107E4)
      ToralRootDatum(n) : RngIntElt -> RootDtm
      Example RootDtm_ToralRootData (H107E5)
      TrivialRootDatum() : -> RootDat

      Constructing Sparse Root Data
            SparseRootDatum(N) : MonStgElt -> RootDtmSprs
            Example RootDtm_SprsRD (H107E6)
            SparseRootDatum(R) : RootDtm -> RootDtmSprs
            RootDatum(R) : RootDtmSprs -> RootDtm
            Example RootDtm_SprsRDsumsub (H107E7)

 
Operations on Root Data
      R1 eq R2 : RootDtm, RootDtm -> BoolElt
      IsIsomorphic(R1, R2) : RootDtm, RootDtm -> BoolElt, [RngIntElt], Map
      IsCartanEquivalent(R1, R2) : RootDtm, RootDtm -> BoolElt, SeqEnum
      IsIsogenous(R1, R2) : RootDtm, RootDtm -> BoolElt, SeqEnum, RootDtm, Map, Map, RootDtm, Map, Map
      Example RootDtm_IsomorphismIsogeny (H107E8)
      CartanName(R) : RootStr -> MonStgElt
      TwistedCartanName(R) : RootDtm -> MonStgElt
      CoxeterDiagram(R) : RootStr ->
      DynkinDiagram(R) : RootStr ->
      CoxeterMatrix(R) : RootStr -> AlgMatElt
      CoxeterGraph(R) : RootStr -> GrphUnd
      CartanMatrix(R) : RootStr -> AlgMatElt
      DynkinDigraph(R) : RootStr -> GrphDir
      Example RootDtm_Diagrams (H107E9)
      GammaAction(R) : RootDtm -> Rec
      GammaRootSpace(R) : RootDtm -> GSetEnum, Map
      GammaOrbitOnRoots(R,r) : RootDtm, RngIntElt -> GSetEnum
      GammaOrbitsOnRoots(R) : RootDtm -> SeqEnum[GSetEnum]
      GammaActionOnSimples(R) : RootDtm -> HomGrp
      OrbitsOnSimples(R) : RootDtm -> SeqEnum[GSetEnum]
      DistinguishedOrbitsOnSimples(R) : RootDtm -> SeqEnum[GSetEnum]
      BaseRing(R) : RootDtm -> RngInt
      Rank(R) : RootStr -> RngIntElt
      RelativeRank(R) : RootDtm -> RngIntElt
      Dimension(R) : RootStr -> RngIntElt
      TwistingDegree(R) : RootDtm -> RngIntElt
      AnisotropicSubdatum(R) : RootDtm -> RootDtm
      Example RootDtm_OperationsForTwistedRootData (H107E10)
      CoxeterGroupOrder(R) : RootStr -> RngIntElt
      GroupOfLieTypeOrder(R, q) : RootDtm, RngElt -> RngIntElt
      GroupOfLieTypeFactoredOrder(R, q) : RootDtm, RngElt -> RngIntElt
      Example RootDtm_GroupOfLieTypeOrder (H107E11)
      FundamentalGroup(R) : RootDtm -> GrpAb, Map
      IsogenyGroup(R) : RootDtm -> GrpAb, Map
      CoisogenyGroup(R) : RootDtm -> GrpAb, Map
      Example RootDtm_IsogenyGroups (H107E12)

 
Properties of Root Data
      IsFinite(R) : RootStr -> BoolElt
      IsIrreducible(R) : RootStr -> BoolElt
      IsAbsolutelyIrreducible(R) : RootStr -> BoolElt
      IsProjectivelyIrreducible(R) : RootStr -> BoolElt
      IsReduced(R) : RootDtm -> BoolElt
      IsSemisimple(R) : RootStr -> BoolElt
      IsCrystallographic(R) : RootStr -> BoolElt
      IsSimplyLaced(R) : RootStr -> BoolElt
      IsAdjoint(R) : RootDtm -> BoolElt
      IsWeaklyAdjoint(R) : RootDtm -> BoolElt
      IsSimplyConnected(R) : RootDtm -> BoolElt
      IsWeaklySimplyConnected(R) : RootDtm -> BoolElt
      Example RootDtm_Properties (H107E13)
      IsReduced(R) : RootStr -> BoolElt
      IsSplit(R) : RootDtm -> BoolElt
      IsTwisted(R) : RootDtm -> BoolElt
      IsQuasisplit(R) : RootDtm -> BoolElt
      IsInner(R) : RootDtm -> BoolElt
      IsAnisotropic(R) : RootDtm -> BoolElt

 
Roots, Coroots and Weights

      Accessing Roots and Coroots
            RootSpace(R) : RootStr -> ModTupFld
            FullRootLattice(R) : RootDtm -> Lat, Map
            RootLattice(R) : RootDtm -> Lat, Map
            Example RootDtm_RtLat (H107E14)
            IsRootSpace(V) : ModTupFld -> BoolElt
            IsInRootSpace(v) : ModTupFldElt -> BoolElt
            RootDatum(V) : ModTupFld -> RootDtm
            Example RootDtm_RtIsSpace (H107E15)
            ZeroRootLattice(R) : RootDtm -> Lat
            RelativeRootSpace(R) : RootDtm -> ModTupFld, Map
            SimpleRoots(R) : RootStr -> Mtrx
            Example RootDtm_BasicOperations (H107E16)
            NumberOfPositiveRoots(R) : RootStr -> RngIntElt
            Roots(R) : RootStr -> (@@)
            PositiveRoots(R) : RootStr -> (@@)
            Root(R, r) : RootStr, RngIntElt -> (@@)
            RootPosition(R, v) : RootStr, . -> (@@)
            BasisChange(R,v) : RootStr, Any -> SeqEnum
            Example RootDtm_RootsCoroots (H107E17)
            IsInRootSpace(R,v) : RootDtm, ModTupFldElt -> BoolElt
            HighestRoot(R) : RootStr -> .
            HighestLongRoot(R) : RootStr -> .
            HighestShortRoot(R) : RootStr -> .
            Example RootDtm_HighestRoots (H107E18)
            RelativeRoots(R) : RootDtm -> SetIndx
            RelativeRootDatum(R) : RootDtm -> RootDtm
            GammaOrbitsRepresentatives(R, delta) : RootDtm, RngIntElt -> SeqEnum
            Example RootDtm_TwoTwistedEsixes (H107E19)
            CoxeterForm(R) : RootDtm -> AlgMatElt

      Reflections
            SimpleReflectionMatrices(R) : RootDtm -> []
            ReflectionMatrices(R) : RootDtm -> []
            ReflectionMatrix(R, r) : RootDtm, RngIntElt -> []
            SimpleReflectionPermutations(R) : RootDtm -> []
            ReflectionPermutations(R) : RootDtm -> []
            ReflectionPermutation(R, r) : RootDtm, RngIntElt -> []
            ReflectionWords(R) : RootDtm -> []
            ReflectionWord(R, r) : RootDtm, RngIntElt -> []
            Example RootDtm_Action (H107E20)

      Operations and Properties for Root and Coroot Indices
            Sum(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
            IsPositive(R, r) : RootStr, RngIntElt -> BoolElt
            IsNegative(R, r) : RootStr, RngIntElt -> BoolElt
            Negative(R, r) : RootStr, RngIntElt -> RngIntElt
            LeftString(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
            RightString(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
            LeftStringLength(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
            RightStringLength(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
            Example RootDtm_RootArithmetic (H107E21)
            RootHeight(R, r) : RootStr, RngIntElt -> RngIntElt
            RootNorms(R) : RootStr -> [RngIntElt]
            RootNorm(R, r) : RootStr, RngIntElt -> RngIntElt
            IsLongRoot(R, r) : RootStr, RngIntElt -> BoolElt
            IsShortRoot(R, r) : RootStr, RngIntElt -> BoolElt
            IsIndivisibleRoot(R, r) : RootStr, RngIntElt -> BoolElt
            Example RootDtm_RootOperations (H107E22)
            RootClosure(R, S) : RootDtm, SetEnum[RngIntElt] -> SetEnum[RngIntElt]
            AdditiveOrder(R) : RootStr -> SeqEnum
            IsAdditiveOrder(R, Q) : RootStr, [RngIntElt] -> BoolElt
            Example RootDtm_AdditiveOrder (H107E23)

      Weights
            WeightLattice(R) : RootDtm -> Lat
            CoweightLattice(R) : RootDtm -> Lat
            FundamentalWeights(R) : RootDtm -> Mtrx
            FundamentalCoweights(R) : RootDtm -> Mtrx
            Example RootDtm_Weights (H107E24)
            IsDominant(R, v) : RootDtm, . -> ModTupFldElt, GrpFPCoxElt
            DominantWeight(R, v) : RootDtm, . -> ModTupFldElt, GrpFPCoxElt
            WeightOrbit(R, v) : RootDtm, . -> @ ModTupFldElt @, [GrpFPCoxElt]
            Example RootDtm_DominantWeights (H107E25)

 
Building Root Data
      sub<R | a> : RootDtm, SetEnum -> RootDtm
      sub<R | s> : RootDtm, SetEnum -> RootDtm
      Example RootDtm_RootSubdata (H107E26)
      R1 subset R2 : RootDtm, RootDtm -> BoolElt, .
      R1 + R2 : RootDtm, RootDtm -> RootDtm
      R1 join R2 : RootDtm, RootDtm -> RootDtm
      Example RootDtm_RootDtmSums (H107E27)
      DirectSumDecomposition(R) : RootDtm -> [], RootDtm, Map
      Example RootDtm_RootDtmDecomp (H107E28)
      Dual(R) : RootDtm -> RootDtm, Map
      SimplyConnectedVersion(R) : RootDtm -> RootDtm, Map
      AdjointVersion(R) : RootDtm -> RootDtm, Map
      IndivisibleSubdatum(R) : RootDtm -> RootDtm
      Radical(R) : RootDtm -> RootDtm
      Example RootDtm_DirectSumDualRadical (H107E29)
      TwistedRootDatum(R) : RootDtm -> RootDtm
      Example RootDtm_DirectSumDualRadical (H107E30)
      UntwistedRootDatum(R) : RootDtm -> RootDtm

 
Morphisms of Root Data
      hom<R -> S | phiX, phiY> : RootDtm, RootDtm, Map, Map -> Map
      hom<R -> S | Q> : RootDtm, RootDtm, [RngIntElt] -> Map
      Morphism(R, S, phiX, phiY) : RootDtm, RootDtm, Map, Map -> Map
      Morphism(R, S, Q) : RootDtm, RootDtm, [RngIntElt] -> Map
      DualMorphism(R, S, phiX, phiY) : RootDtm, RootDtm, Map, Map -> Map
      DualMorphism(R, S, Q) : RootDtm, RootDtm, [RngIntElt] -> Map
      RootImages(phi) : Map -> [RngIntElt]
      RootPermutation(phi) : Map -> GrpPermElt
      IdentityMap(R) : RootDtm -> Map
      Example RootDtm_CreatingRootDataHomomorphisms (H107E31)

 
Constants Associated with Root Data
      ExtraspecialPairs(R) : RootDtm -> SeqEnum
      NumExtraspecialPairs(R) : RootDtm -> SeqEnum
      ExtraspecialPair(R,r) : RootDtm, RngIntElt -> SeqEnum
      ExtraspecialSigns(R) : RootDtm -> []
      LieConstant_p(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
      LieConstant_q(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
      CartanInteger(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
      LieConstant_N(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
      LieConstant_epsilon(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
      LieConstant_M(R, r, s, i) : RootDtm, RngIntElt, RngIntElt, RngIntElt -> RngIntElt
      LieConstant_C(R, i, j, r, s) : RootDtm, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> RngIntElt
      LieConstant_eta(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
      StructureConstants(R) : RootDtm -> RngIntElt
      Example RootDtm_consts (H107E32)

 
Related Structures
      RootSystem(R) : RootDtm -> RootSys
      CoxeterGroup(grpcat, R) : Cat, RootDtm -> grpcat
      CoxeterGroup(R) : RootDtm -> GrpPermCox
      CoxeterGroup(GrpPermCox, R) : Cat, RootDtm -> GrpPermCox
      LieAlgebraHomorphism(phi,k) : Map, Rng -> AlgLie
      LieAlgebra(R, k) : RootDtm, Rng -> AlgLie
      GroupOfLieType(R, k) : RootDtm, Rng -> GrpLie
      GroupOfLieTypeHomomorphism(phi, k) : Map, Rng -> GrpLie
      Example RootDtm_Related (H107E33)

 
Bibliography

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