Definition of a Split Root Datum
Isogeny of Split Reduced 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
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
Wed Apr 24 15:09:57 EST 2013