# Representations of Lie Groups and Algebras

- [Introduction](representations-introduction.md)

  - [Highest Weight Modules](representations-introduction.md#highest-weight-modules)

  - [Toral Elements](representations-introduction.md#toral-elements)

  - [Other Highest Weight Representations](representations-introduction.md#other-highest-weight-representations)

- [Constructing Weight Multisets](representations-constructing.md)

  - [`TrivialLieRepresentationDecomposition(R): RootDtm → LieRepDec`](representations-constructing.md#function-triviallierepresentationdecomposition-rootdtm)

  - [`LieRepresentationDecomposition(R): RootDtm → LieRepDec`](representations-constructing.md#function-lierepresentationdecomposition-rootdtm)

  - [`LieRepresentationDecomposition(R, v): RootDtm, ModTupRngElt → LieRepDec`](representations-constructing.md#function-lierepresentationdecomposition-rootdtm-modtuprngelt)

  - [`LieRepresentationDecomposition(R, v): RootDtm, SeqEnum → LieRepDec`](representations-constructing.md#function-lierepresentationdecomposition-rootdtm-seqenum)

  - [`LieRepresentationDecomposition(R, Wt, Mp): RootDtm, SeqEnum, SeqEnum → LieRepDec`](representations-constructing.md#function-lierepresentationdecomposition-rootdtm-seqenum-seqenum)

  - [`AdjointRepresentationDecomposition(R): RootDtm → LieRepDec`](representations-constructing.md#function-adjointrepresentationdecomposition-rootdtm)

  - [`Example: Adjoint Representation`](representations-constructing.md#example-ex-e0259e)

- [Constructing Representations](representations-constructing-2.md)

  - [Lie Algebras](representations-constructing-2.md#lie-algebras)

    - [`TrivialRepresentation(L): AlgLie → Map`](representations-constructing-2.md#function-trivialrepresentation-alglie)

    - [`AdjointRepresentation(L): AlgLie → Map`](representations-constructing-2.md#function-adjointrepresentation-alglie)

    - [`StandardRepresentation(L): AlgLie → Map`](representations-constructing-2.md#function-standardrepresentation-alglie)

    - [`Example: Standard Representation`](representations-constructing-2.md#example-ex-b368df)

    - [`HighestWeightRepresentation(L, w): AlgLie, [ ] → UserProgram`](representations-constructing-2.md#function-highestweightrepresentation-alglie)

    - [`Example: Highest Weight`](representations-constructing-2.md#example-ex-42d826)

    - [`HighestWeightModule(L, w): AlgLie, SeqEnum → ModTupAlg`](representations-constructing-2.md#function-highestweightmodule-alglie-seqenum)

    - [`TensorProduct(Q): SeqEnum → ModAlg, Map`](representations-constructing-2.md#function-tensorproduct-seqenum)

    - [`SymmetricPower(V, n): ModAlg, RngIntElt → ModAlg, Map`](representations-constructing-2.md#function-symmetricpower-modalg-rngintelt)

    - [`ExteriorPower(V, n): ModAlg, RngIntElt → ModAlg, Map`](representations-constructing-2.md#function-exteriorpower-modalg-rngintelt)

    - [`Example: Lie Modules`](representations-constructing-2.md#example-ex-a477d3)

  - [Groups of Lie Type](representations-constructing-2.md#groups-of-lie-type)

    - [`TrivialRepresentation(G): GrpLie → Map`](representations-constructing-2.md#function-trivialrepresentation-grplie)

    - [`StandardRepresentation(G): GrpLie → Map`](representations-constructing-2.md#function-standardrepresentation-grplie)

    - [`AdjointRepresentation(G): GrpLie → Map, AlgLie`](representations-constructing-2.md#function-adjointrepresentation-grplie)

    - [`LieAlgebra(G): GrpLie → AlgLie, Map`](representations-constructing-2.md#function-liealgebra-grplie)

    - [`HighestWeightRepresentation(G, v): GrpLie, . → Map`](representations-constructing-2.md#function-highestweightrepresentation-grplie)

    - [`Example: Standard Representation`](representations-constructing-2.md#example-ex-75d620)

- [Operations on Weight Multisets](representation-properties.md)

  - [Basic Operations](representation-properties.md#basic-operations)

    - [`RootDatum(D): LieRepDec → RootDtm`](representation-properties.md#function-rootdatum-lierepdec)

    - [`Weights(D): LieRepDec → SeqEnum, SeqEnum`](representation-properties.md#function-weights-lierepdec)

    - [`WeightsAndMultiplicities(D): LieRepDec → SeqEnum, SeqEnum`](representation-properties.md#function-weightsandmultiplicities-lierepdec)

    - [`Multiset(D): LieRepDec → SetMulti`](representation-properties.md#function-multiset-lierepdec)

    - [`Multiplicity(D, v): LieRepDec, ModTupRngElt → RngIntElt`](representation-properties.md#function-multiplicity-lierepdec-modtuprngelt)

    - [`Multiplicity(D, v): LieRepDec, SeqEnum → RngIntElt`](representation-properties.md#function-multiplicity-lierepdec-seqenum)

    - [`D eq E: LieRepDec, LieRepDec → BoolElt`](representation-properties.md#operation-op-eq-lierepdec-lierepdec)

    - [`D + E: LieRepDec, LieRepDec → BoolElt`](representation-properties.md#operation-op-plus-lierepdec-lierepdec)

    - [`D +:= E: LieRepDec, LieRepDec`](representation-properties.md#operation-operation-lierepdec-lierepdec)

    - [`AddRepresentation(~D, E, c): LieRepDec, LieRepDec, RngIntElt`](representation-properties.md#function-addrepresentation-lierepdec-lierepdec-rngintelt-ref)

    - [`AddRepresentation(~D, E): LieRepDec, LieRepDec`](representation-properties.md#function-addrepresentation-lierepdec-lierepdec-ref)

    - [`D + v: LieRepDec, ModTupRngElt → BoolElt`](representation-properties.md#operation-op-plus-lierepdec-modtuprngelt)

    - [`D + v: LieRepDec, SeqEnum → BoolElt`](representation-properties.md#operation-op-plus-lierepdec-seqenum)

    - [`AddRepresentation(~D, v, c): LieRepDec, ModTupRngElt, RngIntElt`](representation-properties.md#function-addrepresentation-lierepdec-modtuprngelt-rngintelt-ref)

    - [`AddRepresentation(~D, v, c): LieRepDec, SeqEnum, RngIntElt`](representation-properties.md#function-addrepresentation-lierepdec-seqenum-rngintelt-ref)

    - [`AddRepresentation(~D, v): LieRepDec, ModTupRngElt`](representation-properties.md#function-addrepresentation-lierepdec-modtuprngelt-ref)

    - [`AddRepresentation(~D, v): LieRepDec, SeqEnum`](representation-properties.md#function-addrepresentation-lierepdec-seqenum-ref)

    - [`D +:= v: LieRepDec, ModTupRngElt`](representation-properties.md#operation-operation-lierepdec-modtuprngelt)

    - [`D +:= v: LieRepDec, SeqEnum`](representation-properties.md#operation-operation-lierepdec-seqenum)

    - [`D * c: LieRepDec, RngIntElt → LieRepDec`](representation-properties.md#operation-op-times-lierepdec-rngintelt)

    - [`D / c: LieRepDec, RngIntElt → LieRepDec`](representation-properties.md#operation-op-div-lierepdec-rngintelt)

    - [`D *:= c: LieRepDec, RngIntElt`](representation-properties.md#operation-operation-lierepdec-rngintelt)

    - [`D /:= c: LieRepDec, RngIntElt`](representation-properties.md#operation-operation-lierepdec-rngintelt-2)

    - [`D * E: LieRepDec, LieRepDec → LieRepDec`](representation-properties.md#operation-op-times-lierepdec-lierepdec)

    - [`ProductRepresentation(D, E): LieRepDec, LieRepDec → LieRepDec`](representation-properties.md#function-productrepresentation-lierepdec-lierepdec)

    - [`ProductRepresentation(D, E, R): LieRepDec, LieRepDec, RootDtm → LieRepDec`](representation-properties.md#function-productrepresentation-lierepdec-lierepdec-rootdtm)

    - [`SubWeights(D, Q, S): LieRepDec, SeqEnum, RootDtm → LieRepDec`](representation-properties.md#function-subweights-lierepdec-seqenum-rootdtm)

    - [`PermuteWeights(D, pi, S): LieRepDec, GrpPermElt, RootDtm → LieRepDec`](representation-properties.md#function-permuteweights-lierepdec-grppermelt-rootdtm)

    - [`Example: Decomp Arithmetic`](representation-properties.md#example-ex-122af0)

  - [Conversion Functions](representation-properties.md#conversion-functions)

    - [`VirtualDecomposition(C): LieRepDec → LieRepDec`](representation-properties.md#function-virtualdecomposition-lierepdec)

    - [`VirtualDecomposition(R, v): RootDtm, Any → LieRepDec`](representation-properties.md#function-virtualdecomposition-rootdtm-any)

    - [`DecomposeCharacter(C): LieRepDec → LieRepDec`](representation-properties.md#function-decomposecharacter-lierepdec)

    - [`DominantCharacter(D): LieRepDec → LieRepDec`](representation-properties.md#function-dominantcharacter-lierepdec)

  - [Calculating with Representations](representation-properties.md#calculating-with-representations)

    - [`RepresentationDimension(D): LieRepDec → RngIntElt`](representation-properties.md#function-representationdimension-lierepdec)

    - [`RepresentationDimension(R, v): RootDtm, SeqEnum → RngIntElt`](representation-properties.md#function-representationdimensionrv)

    - [`RepresentationDimension(R, v): RootDtm, ModTupRngElt → RngIntElt`](representation-properties.md#function-representationdimension-rootdtm-modtuprngelt)

    - [`CasimirValue(R, w): RootDtm, ModTupRngElt → FldRatElt`](representation-properties.md#function-casimirvalue-rootdtm-modtuprngelt)

    - [`CasimirValue(R, w): RootDtm, SeqEnum[RngIntElt] → FldRatElt`](representation-properties.md#function-casimirvalue-rootdtm-seqenum-rngintelt)

    - [`QuantumDimension(R, w): RootDtm, ModTupRngElt → SetMulti`](representation-properties.md#function-quantumdimension-rootdtm-modtuprngelt)

    - [`QuantumDimension(R, w): RootDtm, SeqEnum[RngIntElt] → SetMulti`](representation-properties.md#function-quantumdimension-rootdtm-seqenum-rngintelt)

    - [`Example: Quantum Dimension`](representation-properties.md#example-ex-e2320d)

    - [`Branch(FromGrp, ToGrp, v, M): RootDtm, RootDtm, ModTupRngElt, AlgMatElt → LieRepDec`](representation-properties.md#function-branchfrtovm)

    - [`Branch(FromGrp, ToGrp, v, M): RootDtm, RootDtm, SeqEnum, AlgMatElt → LieRepDec`](representation-properties.md#function-branch-rootdtm-rootdtm-seqenum-algmatelt)

    - [`Branch(ToGrp, D, M): RootDtm, LieRepDec, AlgMatElt → LieRepDec`](representation-properties.md#function-branchtodm)

    - [`Collect(R, D, M): RootDtm, LieRepDec, AlgMatElt → LieRepDec`](representation-properties.md#function-collect)

    - [`Example: Branch Collect`](representation-properties.md#example-ex-e0936d)

    - [`TensorProduct(R, v, w): RootDtm, ModTupRngElt, ModTupRngElt → .`](representation-properties.md#function-tensorrvw)

    - [`TensorProduct(R, v, w): RootDtm, SeqEnum, SeqEnum → .`](representation-properties.md#function-tensorproduct-rootdtm-seqenum-seqenum)

    - [`TensorProduct(D, E): LieRepDec, LieRepDec → .`](representation-properties.md#function-tensorde)

    - [`TensorProduct(Q): [LieRepDec] → LieRepDec`](representation-properties.md#function-tensorde-2)

    - [`TensorPower(R, n, v): RootDtm, RngIntElt, ModTupRngElt → LieRepDec`](representation-properties.md#function-tensorpower-rootdtm-rngintelt-modtuprngelt)

    - [`TensorPower(R, n, v): RootDtm, RngIntElt, SeqEnum → LieRepDec`](representation-properties.md#function-tensorpower-rootdtm-rngintelt-seqenum)

    - [`TensorPower(D, n): LieRepDec, RngIntElt → LieRepDec`](representation-properties.md#function-tensorpower-lierepdec-rngintelt)

    - [`Example: Tensor Power`](representation-properties.md#example-ex-58edc6)

    - [`AdamsOperator(R, n, v): RootDtm, RngIntElt, ModTupRngElt → LieRepDec`](representation-properties.md#function-adamsoperator-rootdtm-rngintelt-modtuprngelt)

    - [`AdamsOperator(R, n, v): RootDtm, RngIntElt, SeqEnum → LieRepDec`](representation-properties.md#function-adamsoperator-rootdtm-rngintelt-seqenum)

    - [`AdamsOperator(D, n): LieRepDec, RngIntElt → LieRepDec`](representation-properties.md#function-adamsoperator-lierepdec-rngintelt)

    - [`SymmetricPower(R, n, v): RootDtm, RngIntElt, ModTupRngElt → LieRepDec`](representation-properties.md#function-symmetricpower-rootdtm-rngintelt-modtuprngelt)

    - [`SymmetricPower(R, n, v): RootDtm, RngIntElt, SeqEnum → LieRepDec`](representation-properties.md#function-symmetricpower-rootdtm-rngintelt-seqenum)

    - [`SymmetricPower(D, n): LieRepDec, RngIntElt → LieRepDec`](representation-properties.md#function-symmetricpower-lierepdec-rngintelt)

    - [`AlternatingPower(R, n, v): RootDtm, RngIntElt, ModTupRngElt → LieRepDec`](representation-properties.md#function-alternatingpower-rootdtm-rngintelt-modtuprngelt)

    - [`AlternatingPower(R, n, v): RootDtm, RngIntElt, SeqEnum → LieRepDec`](representation-properties.md#function-alternatingpower-rootdtm-rngintelt-seqenum)

    - [`AlternatingPower(D, n): LieRepDec, RngIntElt → LieRepDec`](representation-properties.md#function-alternatingpower-lierepdec-rngintelt)

    - [`Plethysm(R, lambda, v): RootDtm, SeqEnum, ModTupRngElt → LieRepDec`](representation-properties.md#function-plethysm-rootdtm-seqenum-modtuprngelt)

    - [`Plethysm(R, lambda, v): RootDtm, SeqEnum, SeqEnum → LieRepDec`](representation-properties.md#function-plethysm-rootdtm-seqenum-seqenum)

    - [`Plethysm(D, lambda): LieRepDec, SeqEnum → LieRepDec`](representation-properties.md#function-plethysm-lierepdec-seqenum)

    - [`Spectrum(R, v, t): RootDtm, ModTupRngElt, SeqEnum → SeqEnum`](representation-properties.md#function-spectrumrvt)

    - [`Spectrum(R, v, t): RootDtm, SeqEnum, SeqEnum → SeqEnum`](representation-properties.md#function-spectrum-rootdtm-seqenum-seqenum)

    - [`Spectrum(D, t): LieRepDec, SeqEnum → SeqEnum`](representation-properties.md#function-spectrumdt)

    - [`Example: Spectrum`](representation-properties.md#example-ex-66b7b0)

    - [`Demazure(R, v, w): RootDtm, ModTupRngElt, GrpPermElt → LieRepDec`](representation-properties.md#function-demazurervw)

    - [`Demazure(R, v, w): RootDtm, SeqEnum, GrpPermElt → LieRepDec`](representation-properties.md#function-demazure-rootdtm-seqenum-grppermelt)

    - [`Demazure(D, w): LieRepDec, GrpPermElt → LieRepDec`](representation-properties.md#function-demazured)

    - [`Demazure(R, v): RootDtm, ModTupRngElt → LieRepDec`](representation-properties.md#function-demazure-rootdtm-modtuprngelt)

    - [`Demazure(R, v): RootDtm, SeqEnum → LieRepDec`](representation-properties.md#function-demazure-rootdtm-seqenum)

    - [`Demazure(D): LieRepDec → LieRepDec`](representation-properties.md#function-demazure-lierepdec)

    - [`Example: Branch Collect`](representation-properties.md#example-ex-02a067)

    - [`LittlewoodRichardsonTensor(p, q): ModTupRngElt, ModTupRngElt → SeqEnum, SeqEnum[RngIntElt]`](representation-properties.md#function-littlewoodrichardsontensor-pq)

    - [`LittlewoodRichardsonTensor(p, q): SeqEnum, SeqEnum → SeqEnum, SeqEnum[RngIntElt]`](representation-properties.md#function-littlewoodrichardsontensor-seqenum-seqenum)

    - [`LittlewoodRichardsonTensor(P, M, Q, N): SeqEnum, SeqEnum[RngIntElt], SeqEnum, SeqEnum[RngIntElt] → SeqEnum, SeqEnum[RngIntElt]`](representation-properties.md#function-littlewoodrichardsontensor-seqenum-seqenum-rngintelt-seqenum-seqenum-rngintelt)

    - [`LittlewoodRichardsonTensor(R, v, w): RootDtm, ModTupRngElt, ModTupRngElt → LieRepDec`](representation-properties.md#function-littlewoodrichardsontensor-rootdtm-modtuprngelt-modtuprngelt)

    - [`LittlewoodRichardsonTensor(R, v, w): RootDtm, SeqEnum, SeqEnum → LieRepDec`](representation-properties.md#function-littlewoodrichardsontensor-rootdtm-seqenum-seqenum)

    - [`LittlewoodRichardsonTensor(D, E): LieRepDec, LieRepDec → LieRepDec`](representation-properties.md#function-littlewoodrichardsontensor-lierepdec-lierepdec)

    - [`Example: LR Tensor`](representation-properties.md#example-ex-32f0be)

    - [`AlternatingDominant(D, w): LieRepDec, GrpPermElt → LieRepDec`](representation-properties.md#function-altdomdw)

    - [`AlternatingDominant(R, wt, w): RootDtm, ModTupRngElt, GrpPermElt → LieRepDec`](representation-properties.md#function-altdomrwtw)

    - [`AlternatingDominant(R, wt, w): RootDtm, SeqEnum, GrpPermElt → LieRepDec`](representation-properties.md#function-alternatingdominant-rootdtm-seqenum-grppermelt)

    - [`AlternatingDominant(D): LieRepDec → LieRepDec`](representation-properties.md#function-alternatingdominant-lierepdec)

    - [`AlternatingDominant(R, wt): RootDtm, ModTupRngElt → LieRepDec`](representation-properties.md#function-alternatingdominant-rootdtm-modtuprngelt)

    - [`AlternatingDominant(R, wt): RootDtm, SeqEnum → LieRepDec`](representation-properties.md#function-alternatingdominant-rootdtm-seqenum)

    - [`Example: Alternating Dominant`](representation-properties.md#example-ex-96a07b)

    - [`AlternatingWeylSum(R, v): RootDtm, ModTupRngElt → LieRepDec`](representation-properties.md#function-alternatingweylsumrv)

    - [`AlternatingWeylSum(R, v): RootDtm, SeqEnum → LieRepDec`](representation-properties.md#function-alternatingweylsum-rootdtm-seqenum)

    - [`AlternatingWeylSum(D): LieRepDec → LieRepDec`](representation-properties.md#function-alternatingweylsumd)

- [Operations on Representations](representations-operations.md)

  - [Lie Algebras](representations-operations.md#lie-algebras)

    - [`CharacterMultiset(V): ModAlg → LieRepDec`](representations-operations.md#function-charactermultiset-modalg)

    - [`CharacterMultiset(ρ): Map → LieRepDec`](representations-operations.md#function-charactermultiset-map)

    - [`Weights(V): ModAlg → SeqEnum, SeqEnum`](representations-operations.md#function-weights-modalg)

    - [`WeightsAndVectors(V): ModAlg → SeqEnum, SeqEnum`](representations-operations.md#function-weightsandvectors-modalg)

    - [`Weights(ρ): Map → [ModTupRngElt]`](representations-operations.md#function-weights-map)

    - [`WeightsAndVectors(ρ): Map → [ModTupRngElt]`](representations-operations.md#function-weightsandvectors-map)

    - [`HighestWeightsAndVectors(V): ModAlg → SeqEnum, SeqEnum`](representations-operations.md#function-highestweightsandvectors-modalg)

    - [`DecompositionMultiset(V): ModAlg → LieRepDec`](representations-operations.md#function-decompositionmultiset-modalg)

    - [`DecompositionMultiset(ρ): Map → LieRepDec`](representations-operations.md#function-decompositionmultiset-map)

    - [`DominantWeights(R, w): RootDtm, [ ] → [ ModTupRngElt ], [ RngIntElt ]`](representations-operations.md#function-dominantweights-rootdtm)

    - [`WeylDimension(R, w): RootDtm, [ ] → RngIntElt`](representations-operations.md#function-weyldimension-rootdtm)

    - [`Example: Dominant Weights`](representations-operations.md#example-ex-c09ca5)

    - [`DecomposeTensorProduct(R, w, x): RootDtm, [ ], [ ] → [ ModTupRngElt ], [ RngIntElt ]`](representations-operations.md#function-decomposetensorproduct-rootdtm)

    - [`DecomposeSymmetricPower(R, n, w): RootDtm, RngIntElt, [ ] → [ ModTupRngElt ], [ RngIn tElt ]`](representations-operations.md#function-decomposesymmetricpower-rootdtm-rngintelt)

    - [`DecomposeExteriorPower(R, n, w): RootDtm, RngIntElt, [ ] → [ ModTupRngElt ], [ RngIn tElt ]`](representations-operations.md#function-decomposeexteriorpower-rootdtm-rngintelt)

    - [`Example: Decompose Tensor`](representations-operations.md#example-ex-06f529)

    - [`DirectSum(U, V): ModAlg, ModAlg → SeqEnum`](representations-operations.md#function-directsum-modalg-modalg)

    - [`DirectSumDecomposition(V): ModAlg → SeqEnum`](representations-operations.md#function-directsumdecomposition-modalg)

    - [`IndecomposableSummands(V): ModAlg → SeqEnum`](representations-operations.md#function-indecomposablesummands-modalg)

    - [`DirectSum(ρ, τ): ModAlg, ModAlg → SeqEnum`](representations-operations.md#function-directsum-modalg-modalg-2)

    - [`DirectSumDecomposition(ρ): Map[AlgLie, AlgMatLie] → SeqEnum`](representations-operations.md#function-directsumdecomposition-map-alglie-algmatlie)

    - [`IndecomposableSummands(ρ): Map[AlgLie, AlgMatLie] → SeqEnum`](representations-operations.md#function-indecomposablesummands-map-alglie-algmatlie)

    - [`TensorProduct(Q): SeqEnum → ModAlg, Map`](representations-operations.md#function-tensorproduct-seqenum-2)

    - [`SymmetricPower(V, n): ModAlg, RngIntElt → ModAlg, Map`](representations-operations.md#function-symmetricpower-modalg-rngintelt-2)

    - [`ExteriorPower(V, n): ModAlg, RngIntElt → ModAlg, Map`](representations-operations.md#function-exteriorpower-modalg-rngintelt-2)

    - [`Example: Lie Modules`](representations-operations.md#example-ex-c153c8)

  - [Groups of Lie Type](representations-operations.md#groups-of-lie-type)

    - [`DirectSum(ρ, τ): ModAlg, ModAlg → SeqEnum`](representations-operations.md#function-directsum-modalg-modalg-3)

    - [`DirectSumDecomposition(ρ): Map[GrpLie, GrpMat] → SeqEnum`](representations-operations.md#function-directsumdecomposition-map-grplie-grpmat)

    - [`IndecomposableSummands(ρ): Map[GrpLie, GrpMat] → SeqEnum`](representations-operations.md#function-indecomposablesummands-map-grplie-grpmat)

    - [`CharacterMultiset(V): ModAlg → LieRepDec`](representations-operations.md#function-charactermultiset-modalg-2)

    - [`CharacterMultiset(ρ): Map → LieRepDec`](representations-operations.md#function-charactermultiset-map-2)

    - [`Weights(ρ): Map → [LatElt], [ModTupRngElt]`](representations-operations.md#function-weights-map-2)

    - [`WeightsAndVectors(ρ): Map → [LatElt], [ModTupRngElt]`](representations-operations.md#function-weightsandvectors-map-2)

    - [`WeightVectors(ρ): Map → [ModTupRngElt]`](representations-operations.md#function-weightvectors-map)

    - [`Weight(ρ, v): Map, ModTupRngElt → LatElt`](representations-operations.md#function-weight-map-modtuprngelt)

    - [`DecompositionMultiset(V): ModAlg → LieRepDec`](representations-operations.md#function-decompositionmultiset-modalg-2)

    - [`DecompositionMultiset(ρ): Map → LieRepDec`](representations-operations.md#function-decompositionmultiset-map-2)

    - [`HighestWeights(ρ): Map → [LatElt], [ModTupRngElt]`](representations-operations.md#function-highestweights-map)

    - [`HighestWeightVectors(ρ): Map → [ModTupRngElt]`](representations-operations.md#function-highestweightvectors-map)

    - [`GeneralisedRowReduction(ρ): GrpLie, Map → Map`](representations-operations.md#function-generalisedrowreduction-grplie-map)

- [Other Functions for Representation Decompositions](representation-other.md)

  - [`FundamentalClosure(R, S): RootDtm, SetEnum → SetEnum`](representation-other.md#function-fundamentalclosure-rootdtm-setenum)

  - [`Closure(R, S): RootDtm, SetEnum → SetEnum`](representation-other.md#function-closure-rootdtm-setenum)

  - [`RestrictionMatrix(R, Q): RootDtm, SeqEnum → AlgMatElt`](representation-other.md#function-restrictionmatrix-rootdtm-seqenum)

  - [`RestrictionMatrix(R, S): RootDtm, RootDtm → AlgMatElt`](representation-other.md#function-lierepdecrestrictionmatrix)

  - [`Example: Restriction Matrix`](representation-other.md#example-ex-017a6f)

  - [`KLPolynomial(x, y): GrpPermElt, GrpPermElt → RngUPolElt`](representation-other.md#function-klpolynomialxy)

  - [`RPolynomial(x, y): GrpPermElt, GrpPermElt → RngUPolElt`](representation-other.md#function-rpolynomialxy)

  - [`Example: KLPoly RPoly`](representation-other.md#example-ex-3fc4eb)

  - [`Exponents(R): RootDtm → SeqEnum`](representation-other.md#function-exponents-rootdtm)

  - [`Example: Exponents`](representation-other.md#example-ex-9b36b7)

  - [`ToLiE(D): LieRepDec → MonStgElt`](representation-other.md#function-tolie-lierepdec)

  - [`FromLiE(R, p): RootDtm, MonStgElt → LieRepDec`](representation-other.md#function-fromlie-rootdtm-monstgelt)

  - [`Example: To From Li E Ex`](representation-other.md#example-ex-a9c639)

  - [Operations Related to the Symmetric Group](representation-other.md#operations-related-to-the-symmetric-group)

    - [`ConjugationClassLength(l): SeqEnum → RngIntElt`](representation-other.md#function-conjugationclasslength-seqenum)

    - [`PartitionToWeight(l): SeqEnum → SeqEnum`](representation-other.md#function-partitiontoweight-seqenum)

    - [`PartitionToWeight(l): ModTupRngElt → ModTupRngElt`](representation-other.md#function-partitiontoweight-modtuprngelt)

    - [`WeightToPartition(v): SeqEnum → SeqEnum`](representation-other.md#function-weighttopartition-seqenum)

    - [`WeightToPartition(v): ModTupRngElt → ModTupRngElt`](representation-other.md#function-weighttopartition-modtuprngelt)

    - [`TransposePartition(l): SeqEnum → SeqEnum`](representation-other.md#function-transposepartition-seqenum)

    - [`TransposePartition(l): ModTupRngElt → ModTupRngElt`](representation-other.md#function-transposepartition-modtuprngelt)

  - [FusionRules](representation-other.md#fusionrules)

    - [`WZWFusion(R, v, w, k): RootDtm, Any, Any, RngIntElt → SetMulti`](representation-other.md#function-wzwfusion-rootdtm-any-any-rngintelt)

    - [`WZWFusion(D, E, k): LieRepDec, LieRepDec, RngIntElt → LieRepDec`](representation-other.md#function-wzwfusion-lierepdec-lierepdec-rngintelt)

    - [`Example: WZW Fusion`](representation-other.md#example-ex-fa246b)

- [Subgroups of Small Rank](representation-small-database.md)

  - [`LiEMaximalSubgroups() → SeqEnum`](representation-small-database.md#function-liemaximalsubgroups)

  - [`MaximalSubgroups(G): MonStgElt → SeqEnum[MonStgElt]`](representation-small-database.md#function-maximalsubgroups-monstgelt)

  - [`RestrictionMatrix(G, H): MonStgElt, MonStgElt → AlgMatElt`](representation-small-database.md#function-restrictionmatrix-monstgelt-monstgelt)

  - [`Example: Subgroup DB`](representation-small-database.md#example-ex-23802f)

- [Subalgebras of su(d)](sub-subsu.md)

  - [`IrreducibleSimpleSubalgebrasOfSU(N): RngIntElt → SeqEnum`](sub-subsu.md#function-irreduciblesimplesubalgebrasofsu)

  - [`IrreducibleSimpleSubalgebraTreeSU(Q, d): SeqEnum[SeqEnum[Tup]], RngIntElt → GrphDir`](sub-subsu.md#function-irreduciblesimplesubalgebratreesu-seqenum-seqenum-tup-rngintelt)

  - [`PrintTreesSU(Q, F): SeqEnum[SeqEnum[Tup]], MonStgElt`](sub-subsu.md#function-printtreessu-seqenum-seqenum-tup-monstgelt)

  - [`Example: Sub SU`](sub-subsu.md#example-ex-f4d461)