Representations of Lie Groups and Algebras
- Introduction
- Constructing Weight Multisets
- Constructing Representations
- Lie Algebras
TrivialRepresentation(L): AlgLie → MapAdjointRepresentation(L): AlgLie → MapStandardRepresentation(L): AlgLie → Map- Example: Standard Representation of a Lie Algebra
HighestWeightRepresentation(L, w): AlgLie, [ ] → UserProgram- Example: Highest Weight Representation of a Lie Algebra
HighestWeightModule(L, w): AlgLie, SeqEnum → ModTupAlgTensorProduct(Q): SeqEnum → ModAlg, MapSymmetricPower(V, n): ModAlg, RngIntElt → ModAlg, MapExteriorPower(V, n): ModAlg, RngIntElt → ModAlg, Map- Example: Tensor Product of Lie Algebra Modules
- Groups of Lie Type
- Operations on Weight Multisets
- Basic Operations
RootDatum(D): LieRepDec → RootDtmWeights(D): LieRepDec → SeqEnum, SeqEnumWeightsAndMultiplicities(D): LieRepDec → SeqEnum, SeqEnumMultiset(D): LieRepDec → SetMultiMultiplicity(D, v): LieRepDec, ModTupRngElt → RngIntEltMultiplicity(D, v): LieRepDec, SeqEnum → RngIntEltD eq E: LieRepDec, LieRepDec → BoolEltD + E: LieRepDec, LieRepDec → BoolEltD +:= E: LieRepDec, LieRepDecAddRepresentation(~D, E, c): LieRepDec, LieRepDec, RngIntEltAddRepresentation(~D, E): LieRepDec, LieRepDecD + v: LieRepDec, ModTupRngElt → BoolEltD + v: LieRepDec, SeqEnum → BoolEltAddRepresentation(~D, v, c): LieRepDec, ModTupRngElt, RngIntEltAddRepresentation(~D, v, c): LieRepDec, SeqEnum, RngIntEltAddRepresentation(~D, v): LieRepDec, ModTupRngEltAddRepresentation(~D, v): LieRepDec, SeqEnumD +:= v: LieRepDec, ModTupRngEltD +:= v: LieRepDec, SeqEnumD * c: LieRepDec, RngIntElt → LieRepDecD / c: LieRepDec, RngIntElt → LieRepDecD *:= c: LieRepDec, RngIntEltD /:= c: LieRepDec, RngIntEltD * E: LieRepDec, LieRepDec → LieRepDecProductRepresentation(D, E): LieRepDec, LieRepDec → LieRepDecProductRepresentation(D, E, R): LieRepDec, LieRepDec, RootDtm → LieRepDecSubWeights(D, Q, S): LieRepDec, SeqEnum, RootDtm → LieRepDecPermuteWeights(D, pi, S): LieRepDec, GrpPermElt, RootDtm → LieRepDec- Example: Arithmetic with Decompositions
- Conversion Functions
- Calculating with Representations
RepresentationDimension(D): LieRepDec → RngIntEltRepresentationDimension(R, v): RootDtm, SeqEnum → RngIntEltRepresentationDimension(R, v): RootDtm, ModTupRngElt → RngIntEltCasimirValue(R, w): RootDtm, ModTupRngElt → FldRatEltCasimirValue(R, w): RootDtm, SeqEnum[RngIntElt] → FldRatEltQuantumDimension(R, w): RootDtm, ModTupRngElt → SetMultiQuantumDimension(R, w): RootDtm, SeqEnum[RngIntElt] → SetMulti- Example: Dimension and Quantum Dimension
Branch(FromGrp, ToGrp, v, M): RootDtm, RootDtm, ModTupRngElt, AlgMatElt → LieRepDecBranch(FromGrp, ToGrp, v, M): RootDtm, RootDtm, SeqEnum, AlgMatElt → LieRepDecBranch(ToGrp, D, M): RootDtm, LieRepDec, AlgMatElt → LieRepDecCollect(R, D, M): RootDtm, LieRepDec, AlgMatElt → LieRepDec- Example: Branch and Collect
TensorProduct(R, v, w): RootDtm, ModTupRngElt, ModTupRngElt → .TensorProduct(R, v, w): RootDtm, SeqEnum, SeqEnum → .TensorProduct(D, E): LieRepDec, LieRepDec → .TensorProduct(Q): [LieRepDec] → LieRepDecTensorPower(R, n, v): RootDtm, RngIntElt, ModTupRngElt → LieRepDecTensorPower(R, n, v): RootDtm, RngIntElt, SeqEnum → LieRepDecTensorPower(D, n): LieRepDec, RngIntElt → LieRepDec- Example: Tensor Powers of Representations
AdamsOperator(R, n, v): RootDtm, RngIntElt, ModTupRngElt → LieRepDecAdamsOperator(R, n, v): RootDtm, RngIntElt, SeqEnum → LieRepDecAdamsOperator(D, n): LieRepDec, RngIntElt → LieRepDecSymmetricPower(R, n, v): RootDtm, RngIntElt, ModTupRngElt → LieRepDecSymmetricPower(R, n, v): RootDtm, RngIntElt, SeqEnum → LieRepDecSymmetricPower(D, n): LieRepDec, RngIntElt → LieRepDecAlternatingPower(R, n, v): RootDtm, RngIntElt, ModTupRngElt → LieRepDecAlternatingPower(R, n, v): RootDtm, RngIntElt, SeqEnum → LieRepDecAlternatingPower(D, n): LieRepDec, RngIntElt → LieRepDecPlethysm(R, lambda, v): RootDtm, SeqEnum, ModTupRngElt → LieRepDecPlethysm(R, lambda, v): RootDtm, SeqEnum, SeqEnum → LieRepDecPlethysm(D, lambda): LieRepDec, SeqEnum → LieRepDecSpectrum(R, v, t): RootDtm, ModTupRngElt, SeqEnum → SeqEnumSpectrum(R, v, t): RootDtm, SeqEnum, SeqEnum → SeqEnumSpectrum(D, t): LieRepDec, SeqEnum → SeqEnum- Example: Spectrum of a Toral Element
Demazure(R, v, w): RootDtm, ModTupRngElt, GrpPermElt → LieRepDecDemazure(R, v, w): RootDtm, SeqEnum, GrpPermElt → LieRepDecDemazure(D, w): LieRepDec, GrpPermElt → LieRepDecDemazure(R, v): RootDtm, ModTupRngElt → LieRepDecDemazure(R, v): RootDtm, SeqEnum → LieRepDecDemazure(D): LieRepDec → LieRepDec- Example: Demazure Operator
LittlewoodRichardsonTensor(p, q): ModTupRngElt, ModTupRngElt → SeqEnum, SeqEnum[RngIntElt]LittlewoodRichardsonTensor(p, q): SeqEnum, SeqEnum → SeqEnum, SeqEnum[RngIntElt]LittlewoodRichardsonTensor(P, M, Q, N): SeqEnum, SeqEnum[RngIntElt], SeqEnum, SeqEnum[RngIntElt] → SeqEnum, SeqEnum[RngIntElt]LittlewoodRichardsonTensor(R, v, w): RootDtm, ModTupRngElt, ModTupRngElt → LieRepDecLittlewoodRichardsonTensor(R, v, w): RootDtm, SeqEnum, SeqEnum → LieRepDecLittlewoodRichardsonTensor(D, E): LieRepDec, LieRepDec → LieRepDec- Example: Littlewood-Richardson Tensor Product
AlternatingDominant(D, w): LieRepDec, GrpPermElt → LieRepDecAlternatingDominant(R, wt, w): RootDtm, ModTupRngElt, GrpPermElt → LieRepDecAlternatingDominant(R, wt, w): RootDtm, SeqEnum, GrpPermElt → LieRepDecAlternatingDominant(D): LieRepDec → LieRepDecAlternatingDominant(R, wt): RootDtm, ModTupRngElt → LieRepDecAlternatingDominant(R, wt): RootDtm, SeqEnum → LieRepDec- Example: Computing the Alternating Dominant
AlternatingWeylSum(R, v): RootDtm, ModTupRngElt → LieRepDecAlternatingWeylSum(R, v): RootDtm, SeqEnum → LieRepDecAlternatingWeylSum(D): LieRepDec → LieRepDec
- Operations on Representations
- Lie Algebras
CharacterMultiset(V): ModAlg → LieRepDecCharacterMultiset(ρ): Map → LieRepDecWeights(V): ModAlg → SeqEnum, SeqEnumWeightsAndVectors(V): ModAlg → SeqEnum, SeqEnumWeights(ρ): Map → [ModTupRngElt]WeightsAndVectors(ρ): Map → [ModTupRngElt]HighestWeightsAndVectors(V): ModAlg → SeqEnum, SeqEnumDecompositionMultiset(V): ModAlg → LieRepDecDecompositionMultiset(ρ): Map → LieRepDecDominantWeights(R, w): RootDtm, [ ] → [ ModTupRngElt ], [ RngIntElt ]WeylDimension(R, w): RootDtm, [ ] → RngIntElt- Example: Dominant Weights and Their Multiplicities
DecomposeTensorProduct(R, w, x): RootDtm, [ ], [ ] → [ ModTupRngElt ], [ RngIntElt ]DecomposeSymmetricPower(R, n, w): RootDtm, RngIntElt, [ ] → [ ModTupRngElt ], [ RngIn tElt ]DecomposeExteriorPower(R, n, w): RootDtm, RngIntElt, [ ] → [ ModTupRngElt ], [ RngIn tElt ]- Example: Decomposing Tensor Products
DirectSum(U, V): ModAlg, ModAlg → SeqEnumDirectSumDecomposition(V): ModAlg → SeqEnumIndecomposableSummands(V): ModAlg → SeqEnumDirectSum(ρ, τ): ModAlg, ModAlg → SeqEnumDirectSumDecomposition(ρ): Map[AlgLie, AlgMatLie] → SeqEnumIndecomposableSummands(ρ): Map[AlgLie, AlgMatLie] → SeqEnumTensorProduct(Q): SeqEnum → ModAlg, MapSymmetricPower(V, n): ModAlg, RngIntElt → ModAlg, MapExteriorPower(V, n): ModAlg, RngIntElt → ModAlg, Map- Example: Tensor Product of Lie Algebra Modules
- Groups of Lie Type
DirectSum(ρ, τ): ModAlg, ModAlg → SeqEnumDirectSumDecomposition(ρ): Map[GrpLie, GrpMat] → SeqEnumIndecomposableSummands(ρ): Map[GrpLie, GrpMat] → SeqEnumCharacterMultiset(V): ModAlg → LieRepDecCharacterMultiset(ρ): Map → LieRepDecWeights(ρ): Map → [LatElt], [ModTupRngElt]WeightsAndVectors(ρ): Map → [LatElt], [ModTupRngElt]WeightVectors(ρ): Map → [ModTupRngElt]Weight(ρ, v): Map, ModTupRngElt → LatEltDecompositionMultiset(V): ModAlg → LieRepDecDecompositionMultiset(ρ): Map → LieRepDecHighestWeights(ρ): Map → [LatElt], [ModTupRngElt]HighestWeightVectors(ρ): Map → [ModTupRngElt]GeneralisedRowReduction(ρ): GrpLie, Map → Map
- Other Functions for Representation Decompositions
FundamentalClosure(R, S): RootDtm, SetEnum → SetEnumClosure(R, S): RootDtm, SetEnum → SetEnumRestrictionMatrix(R, Q): RootDtm, SeqEnum → AlgMatEltRestrictionMatrix(R, S): RootDtm, RootDtm → AlgMatElt- Example: Constructing a Restriction Matrix
KLPolynomial(x, y): GrpPermElt, GrpPermElt → RngUPolEltRPolynomial(x, y): GrpPermElt, GrpPermElt → RngUPolElt- Example: Kazhdan-Lusztig and R-Polynomials
Exponents(R): RootDtm → SeqEnum- Example: Exponents of a Coxeter Group
ToLiE(D): LieRepDec → MonStgEltFromLiE(R, p): RootDtm, MonStgElt → LieRepDec- Example: Conversion to and from LiE
- Operations Related to the Symmetric Group
- FusionRules
WZWFusion(R, v, w, k): RootDtm, Any, Any, RngIntElt → SetMultiWZWFusion(D, E, k): LieRepDec, LieRepDec, RngIntElt → LieRepDec- Example: WZW Fusion Rules
- Subgroups of Small Rank
- Subalgebras of su(d)
- Bibliography