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- Introduction
- Constructing Weight Multisets
- Constructing Representations
- Lie Algebras
- TrivialRepresentation(L) : AlgLie -> Map
- AdjointRepresentation(L) : AlgLie -> Map
- StandardRepresentation(L) : AlgLie -> Map
- Example LieReps_StandardRepresentation (H114E2)
- HighestWeightRepresentation(L, w) : AlgLie, [ ] -> UserProgram
- Example LieReps_HighestWeight (H114E3)
- 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 (H114E4)
- Groups of Lie Type
- 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 (H114E6)
- Conversion Functions
- 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 (H114E7)
- 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 (H114E8)
- 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 (H114E9)
- 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 (H114E10)
- Demazure(R, v, w) : RootDtm, ModTupRngElt, GrpPermElt -> LieRepDec
- Demazure(R, v) : RootDtm, ModTupRngElt -> LieRepDec
- Example LieReps_BranchCollect (H114E11)
- LittlewoodRichardsonTensor(p, q) : ModTupRngElt, ModTupRngElt -> SeqEnum, SeqEnum[RngIntElt]
- Example LieReps_LRTensor (H114E12)
- AlternatingDominant(D, w) : LieRepDec, GrpPermElt -> LieRepDec
- AlternatingDominant(D) : LieRepDec -> LieRepDec
- Example LieReps_AlternatingDominant (H114E13)
- AlternatingWeylSum(R, v) : RootDtm, ModTupRngElt -> LieRepDec
- Operations on Representations
- Lie Algebras
- CharacterMultiset(V) : ModAlg -> LieRepDec
- Weights(V) : ModAlg -> SeqEnum, SeqEnum
- Weights(ρ) : Map -> [ModTupRngElt]
- HighestWeightsAndVectors(V) : ModAlg -> SeqEnum, SeqEnum
- DecompositionMultiset(V) : ModAlg -> LieRepDec
- DominantWeights(R, w) : RootDtm, [ ] -> [ ModTupRngElt ], [ RngIntElt ]
- WeylDimension(R, w) : RootDtm, [ ] -> RngIntElt
- Example LieReps_DominantWeights (H114E14)
- DecomposeTensorProduct(R, w, x) : RootDtm, [ ], [ ] -> [ ModTupRngElt ], [ RngIntElt ]
- DecomposeSymmetricPower(R, n, w) : RootDtm, RngIntElt, [ ] -> [ ModTupRngElt ], [ RngIntElt ]
- DecomposeExteriorPower(R, n, w) : RootDtm, RngIntElt, [ ] -> [ ModTupRngElt ], [ RngIntElt ]
- Example LieReps_DecomposeTensor (H114E15)
- 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 (H114E16)
- Groups of Lie Type
- 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 (H114E17)
- KLPolynomial(x, y) : GrpPermElt, GrpPermElt -> RngUPolElt
- RPolynomial(x, y) : GrpPermElt, GrpPermElt -> RngUPolElt
- Example LieReps_KLPoly_RPoly (H114E18)
- Exponents(R) : RootDtm -> SeqEnum
- Example LieReps_Exponents (H114E19)
- ToLiE(D) : LieRepDec -> MonStgElt
- FromLiE(R, p) : RootDtm, MonStgElt -> LieRepDec
- Example LieReps_ToFromLiEEx (H114E20)
- Operations Related to the Symmetric Group
- FusionRules
- WZWFusion(R, v, w, k) : RootDtm, Any, Any, RngIntElt -> SetMulti
- WZWFusion(D, E, k) : LieRepDec, LieRepDec, RngIntElt -> LieRepDec
- Example LieReps_WZWFusion (H114E21)
- Subgroups of Small Rank
- Subalgebras of su(d)
- Bibliography
V2.29, 21 October 2025