- Introduction
- Constructing Root Data
- 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 (H104E8)
- 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 (H104E9)
- 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 (H104E10)
- CoxeterGroupOrder(R) : RootStr -> RngIntElt
- GroupOfLieTypeOrder(R, q) : RootDtm, RngElt -> RngIntElt
- GroupOfLieTypeFactoredOrder(R, q) : RootDtm, RngElt -> RngIntElt
- Example RootDtm_GroupOfLieTypeOrder (H104E11)
- FundamentalGroup(R) : RootDtm -> GrpAb, Map
- IsogenyGroup(R) : RootDtm -> GrpAb, Map
- CoisogenyGroup(R) : RootDtm -> GrpAb, Map
- Example RootDtm_IsogenyGroups (H104E12)
- Properties of Root Data
- 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 (H104E14)
- IsRootSpace(V) : ModTupFld -> BoolElt
- IsInRootSpace(v) : ModTupFldElt -> BoolElt
- RootDatum(V) : ModTupFld -> RootDtm
- Example RootDtm_RtIsSpace (H104E15)
- ZeroRootLattice(R) : RootDtm -> Lat
- RelativeRootSpace(R) : RootDtm -> ModTupFld, Map
- SimpleRoots(R) : RootStr -> Mtrx
- Example RootDtm_BasicOperations (H104E16)
- 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 (H104E17)
- IsInRootSpace(R,v) : RootDtm, ModTupFldElt -> BoolElt
- HighestRoot(R) : RootStr -> .
- HighestLongRoot(R) : RootStr -> .
- HighestShortRoot(R) : RootStr -> .
- Example RootDtm_HighestRoots (H104E18)
- RelativeRoots(R) : RootDtm -> SetIndx
- RelativeRootDatum(R) : RootDtm -> RootDtm
- GammaOrbitsRepresentatives(R, delta) : RootDtm, RngIntElt -> SeqEnum
- Example RootDtm_TwoTwistedEsixes (H104E19)
- CoxeterForm(R) : RootDtm -> AlgMatElt
- Reflections
- 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 (H104E21)
- 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 (H104E22)
- RootClosure(R, S) : RootDtm, SetEnum[RngIntElt] -> SetEnum[RngIntElt]
- AdditiveOrder(R) : RootStr -> SeqEnum
- IsAdditiveOrder(R, Q) : RootStr, [RngIntElt] -> BoolElt
- Example RootDtm_AdditiveOrder (H104E23)
- Weights
- Building Root Data
- sub<R | a> : RootDtm, SetEnum -> RootDtm
- sub<R | s> : RootDtm, SetEnum -> RootDtm
- Example RootDtm_RootSubdata (H104E26)
- R1 subset R2 : RootDtm, RootDtm -> BoolElt, .
- R1 + R2 : RootDtm, RootDtm -> RootDtm
- R1 join R2 : RootDtm, RootDtm -> RootDtm
- Example RootDtm_RootDtmSums (H104E27)
- DirectSumDecomposition(R) : RootDtm -> [], RootDtm, Map
- Example RootDtm_RootDtmDecomp (H104E28)
- 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 (H104E29)
- TwistedRootDatum(R) : RootDtm -> RootDtm
- Example RootDtm_DirectSumDualRadical (H104E30)
- 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 (H104E31)
- 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 (H104E32)
- 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 (H104E33)
- Bibliography
V2.28, 13 July 2023