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Acknowledgements
Introduction
Highest weight modules
Toral Elements
Other Highest Weight Representations
Constructing Weight Multisets
Constructing Representations
Lie Algebras
Groups of Lie Type
Operations on Weight Multisets
Basic Operations
Conversion Functions
Calculating with Representations
Operations on Representations
Lie Algebras
Groups of Lie Type
Other Functions for Representation Decompositions
Operations Related to the Symmetric Group
Subgroups of Small Rank
Bibliography
DETAILS
Introduction
Highest weight modules
Toral Elements
Other Highest Weight Representations
Constructing Weight Multisets
TrivialLieRepresentationDecomposition(R) : RootDtm -> LieRepDec
LieRepresentationDecomposition(R, v) : RootDtm, ModTupFldElt -> LieRepDec
LieRepresentationDecomposition(R, Wt, Mp) : RootDtm, SeqEnum, SeqEnum -> LieRepDec
AdjointRepresentationDecomposition(R) : RootDtm -> LieRepDec
Example LieReps_LieRepDecAdjointEx (H96E1)
Constructing Representations
Lie Algebras
TrivialRepresentation(L) : AlgLie -> Map
StandardRepresentation(L) : AlgLie -> Map
AdjointRepresentation(L) : AlgLie -> Map
Example LieReps_StandardRepresentation (H96E2)
HighestWeightRepresentation(L, w) : AlgLie, [ ] -> UserProgram
Example LieReps_HighestWeight (H96E3)
HighestWeightModule(L, w) : AlgLie, SeqEnum -> ModTupAlg
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 (H96E4)
Operations on Weight Multisets
Basic Operations
RootDatum(D) : LieRepDec -> RootDtm
Weights(D) : LieRepDec -> SeqEnum, SeqEnum
D eq E : LieRepDec, LieRepDec -> BoolElt
D + E : LieRepDec, LieRepDec -> BoolElt
D +:= E : LieRepDec, LieRepDec ->
AddRepresentation(simD, E, c) : LieRepDec, LieRepDec, RngIntElt ->
D + v : LieRepDec, ModTupRngElt -> BoolElt
AddRepresentation(simD, v, c) : LieRepDec, ModTupFldElt, RngIntElt ->
D +:= v : LieRepDec, ModTupFldElt ->
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_LieRepDecArithmeticEx (H96E5)
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_LieRepDecQuantumDimensionEx (H96E6)
Branch(FromGrp, ToGrp, v, M) : RootDtm, RootDtm,ModTupFldElt, AlgMatElt -> LieRepDec
Branch(ToGrp, D, M) : RootDtm, LieRepDec, AlgMatElt ->LieRepDec
Collect(R, D, M) : RootDtm, LieRepDec, AlgMatElt -> LieRepDec
Example LieReps_LieRepDecBranchCollectEx (H96E7)
TensorProduct(R, v, w) : RootDtm, ModTupFldElt, ModTupFldElt -> .
TensorProduct(Q) : [LieRepDec] -> LieRepDec
TensorPower(R, n, v) : RootDtm, RngIntElt, ModTupFldElt -> LieRepDec
Example LieReps_LieRepDecTensorPowerEx (H96E8)
AdamsOperator(R, n, v) : RootDtm, RngIntElt, ModTupFldElt -> LieRepDec
SymmetricPower(R, n, v) : RootDtm, RngIntElt, ModTupFldElt -> LieRepDec
AlternatingPower(R, n, v) : RootDtm, RngIntElt, ModTupFldElt -> LieRepDec
Plethysm(R, lambda, v) : RootDtm, SeqEnum, ModTupFldElt -> LieRepDec
Spectrum(R, v, t) : RootDtm, ModTupFldElt, SeqEnum ->SeqEnum
Example LieReps_LieRepDecSpectrumEx (H96E9)
Demazure(R, v, w) : RootDtm, ModTupFldElt, GrpPermElt ->LieRepDec
Demazure(R, v) : RootDtm, ModTupFldElt -> LieRepDec
Example LieReps_LieRepDecBranchCollectEx (H96E10)
LittlewoodRichardsonTensor(v, w) : ModTupFldElt, ModTupFldElt -> LieRepDec
Example LieReps_LieRepDecLRTensorEx (H96E11)
AlternatingDominant(D, w) : LieRepDec, GrpPermElt -> LieRepDec
AlternatingDominant(D) : LieRepDec, GrpPermElt -> LieRepDec
Example LieReps_LieRepDecAltDomEx (H96E12)
AlternatingWeylSum(R, v) : RootDtm, ModTupFldElt ->LieRepDec
Operations on Representations
Lie Algebras
CharacterMultiset(V) : ModAlg -> LieRepDec
Weights(V) : ModAlg -> SeqEnum, SeqEnum
Weights(ρ) : Map -> [ModTupRngElt]
DecompositionMultiset(V) : ModAlg -> LieRepDec
HighestWeightsAndVectors(V) : ModAlg -> SeqEnum, SeqEnum
DirectSumDecomposition(V) : ModAlg -> SeqEnum
TensorProduct(Q) : SeqEnum -> ModAlg, Map
SymmetricPower(V, n) : ModAlg, RngIntElt -> ModAlg, Map
ExteriorPower(V, n) : ModAlg, RngIntElt -> ModAlg, Map
Example LieReps_LieModules (H96E13)
Groups of Lie Type
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, S) : RootDtm, RootDtm -> AlgMatElt
Example LieReps_LieRepDecResMatEx (H96E14)
KLPolynomial(x, y) : GrpPermElt, GrpPermElt -> RngUPolElt
RPolynomial(x, y) : GrpPermElt, GrpPermElt -> RngUPolElt
Example LieReps_LieRepDecPolysEx (H96E15)
Exponents(R) : RootDtm -> SeqEnum
Example LieReps_LieRepDecExponentsEx (H96E16)
ToLiE(D) : LieRepDec -> MonStgElt
FromLiE(R, p) : RootDtm, MonStgElt -> LieRepDec
Example LieReps_LieRepDecToFromLiEEx (H96E17)
Operations Related to the Symmetric Group
ConjugationClassLength(l) : SeqEnum -> RngIntElt
PartitionToWeight(l) : SeqEnum -> SeqEnum
WeightToPartition(v) : SeqEnum -> SeqEnum
TransposePartition(l) : SeqEnum -> SeqEnum
Subgroups of Small Rank
LiEMaximalSubgroups() : -> SeqEnum
MaximalSubgroups(G) : MonStgElt -> SeqEnum[MonStgElt]
RestrictionMatrix(G, H) : MonStgElt, MonStgElt -> AlgMatElt
Bibliography
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