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ROOT DATA

 
Acknowledgements
 
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
      Constructing Sparse Root Data
 
Operations on Root Data
 
Properties of Root Data
 
Roots, Coroots and Weights
      Accessing Roots and Coroots
      Reflections
      Operations and Properties for Root and Coroot Indices
      Weights
 
Building Root Data
 
Morphisms of Root Data
 
Constants Associated with Root Data
 
Related Structures
 
Bibliography







DETAILS

 
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 (H97E1)
      Example RootDtm_CreatingExtendedRootData (H97E2)
      RootDatum(C) : AlgMatElt -> RootDtm
      RootDatum(D) : GrphDir -> RootDtm
      RootDatum(A, B) : Mtrx, Mtrx -> RootDtm
      Example RootDtm_G2RootSystem (H97E3)
      IrreducibleRootDatum(X, n) : MonStgElt, RngIntElt -> RootDtm
      StandardRootDatum(X, n) : MonStgElt, RngIntElt -> RootDtm
      Example RootDtm_IrreducibleRootDatum (H97E4)
      ToralRootDatum(n) : RngIntElt -> RootDtm
      Example RootDtm_ToralRootData (H97E5)
      TrivialRootDatum() : -> RootDat

      Constructing Sparse Root Data
            SparseRootDatum(N) : MonStgElt -> RootDtmSprs
            Example RootDtm_SprsRD (H97E6)
            SparseRootDatum(R) : RootDtm -> RootDtmSprs
            RootDatum(R) : RootDtmSprs -> RootDtm
            Example RootDtm_SprsRDsumsub (H97E7)

 
Operations on Root Data
      R1 eq R2 : RootDtm, RootDtm -> BoolElt
      IsIsomorphic(R1, R2) : RootDtm, RootDtm -> BoolElt, [RngIntElt], Map
      IsCartanEquivalent(R1, R2) : RootDtm, RootDtm -> BoolElt
      IsIsogenous(R1, R2) : RootDtm, RootDtm -> BoolElt
      Example RootDtm_IsomorphismIsogeny (H97E8)
      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 (H97E9)
      Example RootDtm_BasicOperations (H97E10)
      GammaAction(R) : RootDtm -> Rec
      GammaRootSpace(R) : RootDtm, RngIntElt -> GSetEnum
      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 (H97E11)
      CoxeterGroupOrder(R) : RootStr -> RngIntElt
      GroupOfLieTypeOrder(R, q) : RootDtm, RngElt -> RngIntElt
      GroupOfLieTypeFactoredOrder(R, q) : RootDtm, RngElt -> RngIntElt
      Example RootDtm_GroupOfLieTypeOrder (H97E12)
      FundamentalGroup(R) : RootDtm -> GrpAb, Map
      IsogenyGroup(R) : RootDtm -> GrpAb, Map
      CoisogenyGroup(R) : RootDtm -> GrpAb, Map
      Example RootDtm_IsogenyGroups (H97E13)

 
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 (H97E14)
      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 (H97E15)
            IsRootSpace(V) : ModTupFld -> BoolElt
            IsInRootSpace(v) : ModTupFldElt -> BoolElt
            RootDatum(V) : ModTupFld -> RootDtm
            Example RootDtm_RtIsSpace (H97E16)
            ZeroRootLattice(R) : RootDtm -> Lat
            RelativeRootSpace(R) : RootDtm -> ModTupFld, Map
            SimpleRoots(R) : RootStr -> Mtrx
            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 (H97E17)
            IsInRootSpace(R,v) : RootDtm, ModTupFldElt -> BoolElt
            HighestRoot(R) : RootStr -> .
            HighestLongRoot(R) : RootStr -> .
            HighestShortRoot(R) : RootStr -> .
            Example RootDtm_HighestRoots (H97E18)
            RelativeRoots(R) : RootDtm -> SetIndx
            RelativeRootDatum(R) : RootDtm -> RootDtm
            GammaOrbitsRepresentatives(R, delta) : RootDtm, RngIntElt -> SeqEnum
            Example RootDtm_TwoTwistedEsixes (H97E19)
            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 (H97E20)

      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 (H97E21)
            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 (H97E22)
            RootClosure(R, S) : RootDtm, SetEnum[RngIntElt] -> SetEnum[RngIntElt]
            AdditiveOrder(R) : RootStr -> SeqEnum
            IsAdditiveOrder(R, Q) : RootStr, [RngIntElt] -> BoolElt
            Example RootDtm_AdditiveOrder (H97E23)

      Weights
            WeightLattice(R) : RootDtm -> Lat
            CoweightLattice(R) : RootDtm -> Lat
            FundamentalWeights(R) : RootDtm -> Mtrx
            FundamentalCoweights(R) : RootDtm -> Mtrx
            Example RootDtm_Weights (H97E24)
            IsDominant(R, v) : RootDtm, . -> ModTupFldElt, GrpFPCoxElt
            DominantWeight(R, v) : RootDtm, . -> ModTupFldElt, GrpFPCoxElt
            WeightOrbit(R, v) : RootDtm, . -> @ ModTupFldElt @, [GrpFPCoxElt]
            Example RootDtm_DominantWeights (H97E25)

 
Building Root Data
      sub<R | a> : RootDtm, SetEnum -> RootDtm
      sub<R | s> : RootDtm, SetEnum -> RootDtm
      Example RootDtm_RootSubdata (H97E26)
      R1 subset R2 : RootDtm, RootDtm -> BoolElt, .
      R1 + R2 : RootDtm, RootDtm -> RootDtm
      R1 join R2 : RootDtm, RootDtm -> RootDtm
      Example RootDtm_RootDtmSums (H97E27)
      DirectSumDecomposition(R) : RootDtm -> [], RootDtm, Map
      Example RootDtm_RootDtmDecomp (H97E28)
      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 (H97E29)
      TwistedRootDatum(R) : RootDtm -> RootDtm
      Example RootDtm_DirectSumDualRadical (H97E30)
      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
      IsIsogeny(phi) : Map -> BoolElt
      IdentityMap(R) : RootDtm -> Map
      Example RootDtm_CreatingRootDataHomomorphisms (H97E31)

 
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 (H97E32)

 
Related Structures
      RootSystem(R) : RootDtm -> RootSys
      CoxeterGroup(GrpFPCox, R) : Cat, RootDtm -> GrpFPCox
      CoxeterGroup(R) : RootDtm -> GrpPermCox
      ReflectionGroup(R) : RootDtm -> GrpMat
      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 (H97E33)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012