Root Data
- Introduction
- Constructing Root Data
- Operations on Root Data
R1 eq R2: RootDtm, RootDtm → BoolEltIsIsomorphic(R1, R2): RootDtm, RootDtm → BoolElt, [RngIntElt], MapIsCartanEquivalent(R1, R2): RootDtm, RootDtm → BoolElt, SeqEnumIsIsogenous(R1, R2): RootDtm, RootDtm → BoolElt, SeqEnum, RootDtm, Map, Map, RootDtm, Map, Map- Example: Isomorphism and Isogeny of Root Data
CartanName(R): RootStr → MonStgEltTwistedCartanName(R): RootDtm → MonStgEltCoxeterDiagram(R): RootStrDynkinDiagram(R): RootStrCoxeterMatrix(R): RootStr → AlgMatEltCoxeterGraph(R): RootStr → GrphUndCartanMatrix(R): RootStr → AlgMatEltDynkinDigraph(R): RootStr → GrphDir- Example: Dynkin and Coxeter Diagrams
GammaAction(R): RootDtm → RecGammaRootSpace(R): RootDtm → GSetEnum, MapGammaCorootSpace(R): RootDtm → GSetEnum, MapGammaOrbitOnRoots(R,r): RootDtm, RngIntElt → GSetEnumGammaOrbitsOnRoots(R): RootDtm → SeqEnum[GSetEnum]PositiveGammaOrbitsOnRoots(R): RootDtm → SeqEnum[GSetEnum]NegativeGammaOrbitsOnRoots(R): RootDtm → SeqEnum[GSetEnum]ZeroGammaOrbitsOnRoots(R): RootDtm → SeqEnum[GSetEnum]GammaActionOnSimples(R): RootDtm → HomGrpOrbitsOnSimples(R): RootDtm → SeqEnum[GSetEnum]DistinguishedOrbitsOnSimples(R): RootDtm → SeqEnum[GSetEnum]BaseRing(R): RootDtm → RngIntRank(R): RootStr → RngIntEltAbsoluteRank(R): RootDtm → RngIntEltRelativeRank(R): RootDtm → RngIntEltDimension(R): RootStr → RngIntEltTwistingDegree(R): RootDtm → RngIntEltAnisotropicSubdatum(R): RootDtm → RootDtm- Example: Operations for Twisted Root Data
CoxeterGroupOrder(R): RootStr → RngIntEltGroupOfLieTypeOrder(R, q): RootDtm, RngElt → RngIntEltGroupOfLieTypeFactoredOrder(R, q): RootDtm, RngElt → RngIntElt- Example: Order of a Group of Lie Type
FundamentalGroup(R): RootDtm → GrpAb, MapIsogenyGroup(R): RootDtm → GrpAb, MapCoisogenyGroup(R): RootDtm → GrpAb, Map- Example: Isogeny and Fundamental Groups
- Properties of Root Data
- Roots, Coroots, and Weights
- Accessing Roots and Coroots
RootSpace(R): RootStr → ModTupFldCorootSpace(R): RootStr → ModTupFldFullRootLattice(R): RootDtm → Lat, MapFullCorootLattice(R): RootDtm → Lat, MapRootLattice(R): RootDtm → Lat, MapCorootLattice(R): RootDtm → Lat, Map- Example: Root Space and Root Lattices
IsRootSpace(V): ModTupFld → BoolEltIsCorootSpace(V): ModTupFld → BoolEltIsInRootSpace(v): ModTupFldElt → BoolEltIsCorootSpace(v): ModTupFldElt → BoolEltRootDatum(V): ModTupFld → RootDtm- Example: Recognising a Root Space
ZeroRootLattice(R): RootDtm → LatZeroRootSpace(R): RootDtm → ModTupFld, MapRelativeRootSpace(R): RootDtm → ModTupFld, MapSimpleRoots(R): RootStr → MtrxSimpleCoroots(R): RootStr → Mtrx- Example: Spaces and Lattices of a G2 Root Datum
NumberOfPositiveRoots(R): RootStr → RngIntEltNumPosRoots(R): RootStr → RngIntEltRoots(R): RootStr → SetIndxCoroots(R): RootStr → SetIndxPositiveRoots(R): RootStr → SetIndxPositiveCoroots(R): RootStr → SetIndxRoot(R, r): RootStr, RngIntElt → SetIndxCoroot(R, r): RootStr, RngIntElt → SetIndxRootPosition(R, v): RootStr, . → SetIndxCorootPosition(R, v): RootStr, . → SetIndxBasisChange(R,v): RootStr, Any → SeqEnum- Example: Roots and Coroots of a Root Datum
IsInRootSpace(R,v): RootDtm, ModTupFldElt → BoolEltIsInCorootSpace(R,v): RootDtm, ModTupFldElt → BoolEltHighestRoot(R): RootStr → .HighestCoroot(R): RootStr → .HighestLongRoot(R): RootStr → .HighestLongCoroot(R): RootStr → .HighestShortRoot(R): RootStr → .HighestShortCoroot(R): RootStr → .- Example: Highest Roots
RelativeRoots(R): RootDtm → SetIndxPositiveRelativeRoots(R): RootDtm → SetIndxNegativeRelativeRoots(R): RootDtm → SetIndxSimpleRelativeRoots(R): RootDtm → SetIndxRelativeRootDatum(R): RootDtm → RootDtmGammaOrbitsRepresentatives(R, delta): RootDtm, RngIntElt → SeqEnum- Example: Two Twisted Root Data of Type E6
CoxeterForm(R): RootDtm → AlgMatEltDualCoxeterForm(R): RootDtm → AlgMatElt
- Reflections
SimpleReflectionMatrices(R): RootDtm → []SimpleCoreflectionMatrices(R): RootDtm → []ReflectionMatrices(R): RootDtm → []CoreflectionMatrices(R): RootDtm → []ReflectionMatrix(R, r): RootDtm, RngIntElt → []CoreflectionMatrix(R, r): RootDtm, RngIntElt → []SimpleReflectionPermutations(R): RootDtm → []ReflectionPermutations(R): RootDtm → []ReflectionPermutation(R, r): RootDtm, RngIntElt → []ReflectionWords(R): RootDtm → []ReflectionWord(R, r): RootDtm, RngIntElt → []- Example: Reflection Matrices, Permutations, and Words
- Operations and Properties for Root and Coroot Indices
Sum(R, r, s): RootDtm, RngIntElt, RngIntElt → RngIntEltIsPositive(R, r): RootStr, RngIntElt → BoolEltIsNegative(R, r): RootStr, RngIntElt → BoolEltNegative(R, r): RootStr, RngIntElt → RngIntEltLeftString(R, r, s): RootDtm, RngIntElt, RngIntElt → RngIntEltRightString(R, r, s): RootDtm, RngIntElt, RngIntElt → RngIntEltLeftStringLength(R, r, s): RootDtm, RngIntElt, RngIntElt → RngIntEltRightStringLength(R, r, s): RootDtm, RngIntElt, RngIntElt → RngIntElt- Example: Arithmetic with Root Indices
RootHeight(R, r): RootStr, RngIntElt → RngIntEltCorootHeight(R, r): RootStr, RngIntElt → RngIntEltRootNorms(R): RootStr → [RngIntElt]CorootNorms(R): RootStr → [RngIntElt]RootNorm(R, r): RootStr, RngIntElt → RngIntEltCorootNorm(R, r): RootStr, RngIntElt → RngIntEltIsLongRoot(R, r): RootStr, RngIntElt → BoolEltIsShortRoot(R, r): RootStr, RngIntElt → BoolEltIsIndivisibleRoot(R, r): RootStr, RngIntElt → BoolElt- Example: Heights, Norms, and Root Strings
RootClosure(R, S): RootDtm, SetEnum[RngIntElt] → SetEnum[RngIntElt]AdditiveOrder(R): RootStr → SeqEnumIsAdditiveOrder(R, Q): RootStr, [RngIntElt] → BoolElt- Example: Additive Order on Positive Roots
- Weights
- Building Root Data
sub<R | a>: RootDtm, SetEnum → RootDtmsub<R | s>: RootDtm, SetEnum → RootDtm- Example: Root Subdata
R1 subset R2: RootDtm, RootDtm → BoolElt, .R1 + R2: RootDtm, RootDtm → RootDtmDirectSum(R1, R2): RootDtm, RootDtm → RootDtmR1 join R2: RootDtm, RootDtm → RootDtm- Example: Sums of Root Data
DirectSumDecomposition(R): RootDtm → [], RootDtm, MapIndecomposableSummands(R): RootDtm → [], RootDtm, Map- Example: Decomposing a Root Datum
Dual(R): RootDtm → RootDtm, MapSimplyConnectedVersion(R): RootDtm → RootDtm, MapAdjointVersion(R): RootDtm → RootDtm, MapIndivisibleSubdatum(R): RootDtm → RootDtmRadical(R): RootDtm → RootDtm- Example: Direct Sums, Duals, and Radicals
TwistedRootDatum(R): RootDtm → RootDtmTwistedRootDatum(N): MonStgElt → RootDtm- Example: Constructing Twisted Root Data
UntwistedRootDatum(R): RootDtm → RootDtmSplitRootDatum(R): RootDtm → RootDtm
- Morphisms of Root Data
hom<R -\>S | phiX, phiY>: RootDtm, RootDtm, Map, Map → Maphom<R -\>S | phiX, phiY>: RootDtm, RootDtm, Mtrx, Mtrx → Maphom<R -\>S | Q>: RootDtm, RootDtm, [RngIntElt] → MapMorphism(R, S, phiX, phiY): RootDtm, RootDtm, Map, Map → MapMorphism(R, S, phiX, phiY): RootDtm, RootDtm, Mtrx, Mtrx → MapMorphism(R, S, Q): RootDtm, RootDtm, [RngIntElt] → MapDualMorphism(R, S, phiX, phiY): RootDtm, RootDtm, Map, Map → MapDualMorphism(R, S, phiX, phiY): RootDtm, RootDtm, Mtrx, Mtrx → MapDualMorphism(R, S, Q): RootDtm, RootDtm, [RngIntElt] → MapRootImages(phi): Map → [RngIntElt]RootPermutation(phi): Map → GrpPermEltIdentityMap(R): RootDtm → MapIdentityAutomorphism(R): RootDtm → Map- Example: Creating Root Data Homomorphisms
- Constants Associated with Root Data
ExtraspecialPairs(R): RootDtm → SeqEnumNumExtraspecialPairs(R): RootDtm → SeqEnumExtraspecialPair(R,r): RootDtm, RngIntElt → SeqEnumExtraspecialSigns(R): RootDtm → []LieConstant_p(R, r, s): RootDtm, RngIntElt, RngIntElt → RngIntEltLieConstant_q(R, r, s): RootDtm, RngIntElt, RngIntElt → RngIntEltCartanInteger(R, r, s): RootDtm, RngIntElt, RngIntElt → RngIntEltLieConstant_N(R, r, s): RootDtm, RngIntElt, RngIntElt → RngIntEltLieConstant_epsilon(R, r, s): RootDtm, RngIntElt, RngIntElt → RngIntEltLieConstant_M(R, r, s, i): RootDtm, RngIntElt, RngIntElt, RngIntElt → RngIntEltLieConstant_C(R, i, j, r, s): RootDtm, RngIntElt, RngIntElt, RngIntElt, RngIntElt → RngIntEltLieConstant_eta(R, r, s): RootDtm, RngIntElt, RngIntElt → RngIntEltStructureConstants(R): RootDtm → RngIntElt- Example: Verifying Root Datum Constants
- Related Structures
RootSystem(R): RootDtm → RootSysCoxeterGroup(grpcat, R): Cat, RootDtm → grpcatCoxeterGroup(R): RootDtm → GrpPermCoxWeylGroup(R): RootDtm → GrpPermCoxCoxeterGroup(GrpPermCox, R): Cat, RootDtm → GrpPermCoxReflectionGroup(R): RootDtm → GrpMatLieAlgebraHomorphism(phi,k): Map, Rng → AlgLieLieAlgebra(R, k): RootDtm, Rng → AlgLieGroupOfLieType(R, k): RootDtm, Rng → GrpLieGroupOfLieTypeHomomorphism(phi, k): Map, Rng → GrpLie- Example: Structures Related to a Root Datum
- Bibliography