Other Highest Weight Representations
Constructing Weight Multisets
TrivialLieRepresentationDecomposition(R) : RootDtm -> LieRepDec
LieRepresentationDecomposition(R, v) : RootDtm, ModTupRngElt -> LieRepDec
LieRepresentationDecomposition(R, Wt, Mp) : RootDtm, SeqEnum, SeqEnum -> LieRepDec
AdjointRepresentationDecomposition(R) : RootDtm -> LieRepDec
Example LieReps_AdjointRepresentation (H104E1)
Lie Algebras
TrivialRepresentation(L) : AlgLie -> Map
AdjointRepresentation(L) : AlgLie -> Map
StandardRepresentation(L) : AlgLie -> Map
Example LieReps_StandardRepresentation (H104E2)
HighestWeightRepresentation(L, w) : AlgLie, [ ] -> UserProgram
Example LieReps_HighestWeight (H104E3)
HighestWeightModule(L, w) : AlgLie, SeqEnum -> ModTupAlg
TensorProduct(Q) : SeqEnum -> ModAlg, Map
SymmetricPower(V, n) : ModAlg, RngIntElt -> ModAlg, Map
ExteriorPower(V, n) : ModAlg, RngIntElt -> ModAlg, Map
Example LieReps_LieModules (H104E4)
Groups of Lie Type
TrivialRepresentation(G) : GrpLie -> Map
StandardRepresentation(G) : GrpLie -> Map
AdjointRepresentation(G) : GrpLie -> Map, AlgLie
LieAlgebra(G) : GrpLie -> AlgLie, Map
HighestWeightRepresentation(G, v) : GrpLie, . -> Map
Example LieReps_StandardRepresentation (H104E5)
Operations on Weight Multisets
Basic Operations
RootDatum(D) : LieRepDec -> RootDtm
Weights(D) : LieRepDec -> SeqEnum, SeqEnum
Multiset(D) : LieRepDec -> SetMulti
Multiplicity(D, v) : LieRepDec, ModTupRngElt -> RngIntElt
D eq E : LieRepDec, LieRepDec -> BoolElt
D + E : LieRepDec, LieRepDec -> BoolElt
D +:= E : LieRepDec, LieRepDec ->
AddRepresentation(~D, E, c) : LieRepDec, LieRepDec, RngIntElt ->
D + v : LieRepDec, ModTupRngElt -> BoolElt
AddRepresentation(~D, v, c) : LieRepDec, ModTupRngElt, RngIntElt ->
D +:= v : LieRepDec, ModTupRngElt ->
D * c : LieRepDec, RngIntElt -> LieRepDec
D / c : LieRepDec, RngIntElt -> LieRepDec
D *:= c : LieRepDec, RngIntElt ->
D /:= c : LieRepDec, RngIntElt ->
D * E : LieRepDec, LieRepDec -> LieRepDec
ProductRepresentation(D, E, R) : LieRepDec, LieRepDec, RootDtm -> LieRepDec
SubWeights(D, Q, S) : LieRepDec, SeqEnum, RootDtm -> LieRepDec
PermuteWeights(D, pi, S) : LieRepDec, GrpPermElt, RootDtm -> LieRepDec
Example LieReps_DecompArithmetic (H104E6)
Conversion Functions
VirtualDecomposition(C) : LieRepDec -> LieRepDec
DecomposeCharacter(C) : LieRepDec -> LieRepDec
DominantCharacter(D) : LieRepDec -> LieRepDec
Calculating with Representations
RepresentationDimension(D) : LieRepDec -> RngIntElt
RepresentationDimension(R, v) : RootDtm, SeqEnum -> RngIntElt
CasimirValue(R, w) : RootDtm, ModTupRngElt -> FldRatElt
QuantumDimension(R, w) : RootDtm, ModTupRngElt -> SetMulti
Example LieReps_QuantumDimension (H104E7)
Branch(FromGrp, ToGrp, v, M) : RootDtm, RootDtm, ModTupRngElt, AlgMatElt -> LieRepDec
Branch(ToGrp, D, M) : RootDtm, LieRepDec, AlgMatElt -> LieRepDec
Collect(R, D, M) : RootDtm, LieRepDec, AlgMatElt -> LieRepDec
Example LieReps_BranchCollect (H104E8)
TensorProduct(R, v, w) : RootDtm, ModTupRngElt, ModTupRngElt -> .
TensorProduct(D, E) : LieRepDec, LieRepDec -> .
TensorProduct(Q) : [LieRepDec] -> LieRepDec
TensorPower(R, n, v) : RootDtm, RngIntElt, ModTupRngElt -> LieRepDec
Example LieReps_TensorPower (H104E9)
AdamsOperator(R, n, v) : RootDtm, RngIntElt, ModTupRngElt -> LieRepDec
SymmetricPower(R, n, v) : RootDtm, RngIntElt, ModTupRngElt -> LieRepDec
AlternatingPower(R, n, v) : RootDtm, RngIntElt, ModTupRngElt -> LieRepDec
Plethysm(R, lambda, v) : RootDtm, SeqEnum, ModTupRngElt -> LieRepDec
Spectrum(R, v, t) : RootDtm, ModTupRngElt, SeqEnum -> SeqEnum
Example LieReps_Spectrum (H104E10)
Demazure(R, v, w) : RootDtm, ModTupRngElt, GrpPermElt -> LieRepDec
Demazure(R, v) : RootDtm, ModTupRngElt -> LieRepDec
Example LieReps_BranchCollect (H104E11)
LittlewoodRichardsonTensor(p, q) : ModTupRngElt, ModTupRngElt -> SeqEnum, SeqEnum[RngIntElt]
Example LieReps_LRTensor (H104E12)
AlternatingDominant(D, w) : LieRepDec, GrpPermElt -> LieRepDec
AlternatingDominant(D) : LieRepDec, GrpPermElt -> LieRepDec
Example LieReps_AlternatingDominant (H104E13)
AlternatingWeylSum(R, v) : RootDtm, ModTupRngElt -> LieRepDec
Lie Algebras
CharacterMultiset(V) : ModAlg -> LieRepDec
Weights(V) : ModAlg -> SeqEnum, SeqEnum
Weights(ρ) : Map -> [ModTupRngElt]
DecompositionMultiset(V) : ModAlg -> LieRepDec
HighestWeightsAndVectors(V) : ModAlg -> SeqEnum, SeqEnum
DirectSum(U, V) : ModAlg, ModAlg -> SeqEnum
DirectSumDecomposition(V) : ModAlg -> SeqEnum
DirectSum(ρ, τ) : ModAlg, ModAlg -> SeqEnum
DirectSumDecomposition(ρ) : Map[AlgLie, AlgMatLie] -> SeqEnum
TensorProduct(Q) : SeqEnum -> ModAlg, Map
SymmetricPower(V, n) : ModAlg, RngIntElt -> ModAlg, Map
ExteriorPower(V, n) : ModAlg, RngIntElt -> ModAlg, Map
Example LieReps_LieModules (H104E14)
Groups of Lie Type
DirectSum(ρ, τ) : ModAlg, ModAlg -> SeqEnum
DirectSumDecomposition(ρ) : Map[GrpLie, GrpMat] -> SeqEnum
CharacterMultiset(V) : ModAlg -> LieRepDec
Weights(ρ) : Map -> [LatElt], [ModTupRngElt]
WeightVectors(ρ) : Map -> [ModTupRngElt]
Weight(ρ, v) : Map, ModTupRngElt -> LatElt
DecompositionMultiset(V) : ModAlg -> LieRepDec
HighestWeights(ρ) : Map -> [LatElt], [ModTupRngElt]
HighestWeightVectors(ρ) : Map -> [ModTupRngElt]
GeneralisedRowReduction(ρ) : GrpLie, Map -> Map
Other Functions for Representation Decompositions
FundamentalClosure(R, S) : RootDtm, SetEnum -> SetEnum
Closure(R, S) : RootDtm, SetEnum -> SetEnum
RestrictionMatrix(R, Q) : RootDtm, SeqEnum -> AlgMatElt
RestrictionMatrix(R, S) : RootDtm, RootDtm -> AlgMatElt
Example LieReps_RestrictionMatrix (H104E15)
KLPolynomial(x, y) : GrpPermElt, GrpPermElt -> RngUPolElt
RPolynomial(x, y) : GrpPermElt, GrpPermElt -> RngUPolElt
Example LieReps_KLPoly_RPoly (H104E16)
Exponents(R) : RootDtm -> SeqEnum
Example LieReps_Exponents (H104E17)
ToLiE(D) : LieRepDec -> MonStgElt
FromLiE(R, p) : RootDtm, MonStgElt -> LieRepDec
Example LieReps_ToFromLiEEx (H104E18)
Operations Related to the Symmetric Group
ConjugationClassLength(l) : SeqEnum -> RngIntElt
PartitionToWeight(l) : SeqEnum -> SeqEnum
WeightToPartition(v) : SeqEnum -> SeqEnum
TransposePartition(l) : SeqEnum -> SeqEnum
FusionRules
WZWFusion(R, v, w, k) : RootDtm, Any, Any, RngIntElt -> SetMulti
WZWFusion(D, E, k) : LieRepDec, LieRepDec -> LieRepDec
Example LieReps_WZWFusion (H104E19)
Subgroups of Small Rank
LiEMaximalSubgroups() : -> SeqEnum
MaximalSubgroups(G) : MonStgElt -> SeqEnum[MonStgElt]
RestrictionMatrix(G, H) : MonStgElt, MonStgElt -> AlgMatElt
Example LieReps_SubgroupDB (H104E20)
Subalgebras of su(d)
IrreducibleSimpleSubalgebrasOfSU(N) : RngIntElt -> SeqEnum
IrreducibleSimpleSubalgebraTreeSU(Q, d) : SeqEnum[SeqEnum[Tup]], RngIntElt -> GrphDir
PrintTreesSU(Q, F) : SeqEnum[SeqEnum[Tup]], MonStgElt ->
Example LieReps_SubSU (H104E21)
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012