LieAlgebraRepresentations : Index

• ω -- construct the irreducible Lie algebra module with given highest weight
• 𝔞 (missing documentation)
• 𝔟 (missing documentation)
• 𝔠 (missing documentation)
• 𝔡 (missing documentation)
• 𝔢 (missing documentation)
• 𝔣 (missing documentation)
• 𝔤 (missing documentation)
• adams -- Computes the action of the nth Adams operator on a Lie algebra module
• adams(ZZ,LieAlgebraModule) -- Computes the action of the nth Adams operator on a Lie algebra module
• adjointModule -- The adjoint module of a Lie algebra
• adjointModule(LieAlgebra) -- The adjoint module of a Lie algebra
• adjointRepresentation -- creates the adjoint representation of a Lie algebra
• adjointRepresentation(LieAlgebra) -- creates the adjoint representation of a Lie algebra
• adjointRepresentation(LieAlgebraBasis) -- creates the adjoint representation of a Lie algebra
• adjointRepresentation(String,ZZ) -- creates the adjoint representation of a Lie algebra
• basisWordsFromMatrixGenerators -- express each basis element of $V(\lambda)$ as a linear combination of words in the lowering operators applied to the highest weight vector
• basisWordsFromMatrixGenerators(LieAlgebraRepresentation) -- express each basis element of $V(\lambda)$ as a linear combination of words in the lowering operators applied to the highest weight vector
• branchingRule -- A Lie algebra module viewed as a module over a Lie subalgebra
• branchingRule(LieAlgebraModule,LieAlgebra) -- A Lie algebra module viewed as a module over a Lie subalgebra
• branchingRule(LieAlgebraModule,List) -- A Lie algebra module viewed as a module over a Lie subalgebra
• branchingRule(LieAlgebraModule,Matrix) -- A Lie algebra module viewed as a module over a Lie subalgebra
• branchingRule(LieAlgebraModule,String) -- A Lie algebra module viewed as a module over a Lie subalgebra
• cartanMatrix -- Provide the Cartan matrix of a simple Lie algebra
• cartanMatrix(LieAlgebra) -- Provide the Cartan matrix of a simple Lie algebra
• casimirOperator -- computes the Casimir operator associated to a representation
• casimirOperator(LieAlgebraRepresentation) -- computes the Casimir operator associated to a representation
• casimirProjection -- projection operator to a specified eigenspace of the Casimir operator
• casimirProjection(LieAlgebraRepresentation,QQ) -- projection operator to a specified eigenspace of the Casimir operator
• casimirScalar -- computes the scalar by which the Casimir operator acts on an irreducible Lie algebra module
• casimirScalar(LieAlgebraModule) -- computes the scalar by which the Casimir operator acts on an irreducible Lie algebra module
• casimirSpectrum -- computes the eigenvalues of the Casimir operator associated to a representation
• casimirSpectrum(LieAlgebraModule) -- computes the eigenvalues of the Casimir operator associated to a representation
• character -- Computes the character of a Lie algebra module
• character(...,Strategy=>...) -- Computes the character of a Lie algebra module
• character(LieAlgebra,List) -- Computes the character of a Lie algebra module
• character(LieAlgebra,Vector) -- Computes the character of a Lie algebra module
• character(LieAlgebraModule) -- Computes the character of a Lie algebra module
• characterRing (missing documentation)
• deGraafBases -- compute the bases produced by de Graaf's algorithm
• deGraafBases(List,LieAlgebra) -- compute the bases produced by de Graaf's algorithm
• deGraafRepresentation -- compute the representation with the specified highest weight using de Graaf's algorithm
• deGraafRepresentation(List,LieAlgebra) -- compute the representation with the specified highest weight using de Graaf's algorithm
• dim(LieAlgebraModule) -- computes the dimension of a Lie algebra module as a vector space over the ground field
• directSum(LieAlgebra) -- Take the direct sum of Lie algebras
• directSum(LieAlgebraModule) -- direct sum of LieAlgebraModules
• dual(LieAlgebraModule) -- computes w* for a weight w
• dualCoxeterNumber -- the dual Coxeter number of a simple Lie algebra
• dualCoxeterNumber(LieAlgebra) -- the dual Coxeter number of a simple Lie algebra
• dualCoxeterNumber(String,ZZ) -- the dual Coxeter number of a simple Lie algebra
• dynkinDiagram -- Provide the Dynkin diagram of a simple Lie algebra
• dynkinDiagram(LieAlgebra) -- Provide the Dynkin diagram of a simple Lie algebra
• dynkinToPartition -- converts a highest weight written in the basis of fundamental dominant weights for type A into a partition
• dynkinToPartition(String,List) -- converts a highest weight written in the basis of fundamental dominant weights for type A into a partition
• embedding -- gives the embedding of Cartan subalgebras of one Lie algebra into another
• embedding(LieAlgebra,LieAlgebra) -- gives the embedding of Cartan subalgebras of one Lie algebra into another
• exteriorPower(...,Strategy=>...) -- computes the explicit action on $\bigwedge^k V$ for a $\mathfrak{g}$-module $V$
• exteriorPower(ZZ,LieAlgebraModule) -- Computes the nth symmetric / exterior tensor power of a Lie algebra module
• exteriorPower(ZZ,LieAlgebraRepresentation) -- computes the explicit action on $\bigwedge^k V$ for a $\mathfrak{g}$-module $V$
• fusionCoefficient -- computes the multiplicity of W in the fusion product of U and V
• fusionCoefficient(LieAlgebraModule,LieAlgebraModule,LieAlgebraModule,ZZ) -- computes the multiplicity of W in the fusion product of U and V
• fusionProduct -- computes the multiplicities of irreducibles in the decomposition of the fusion product of U and V
• fusionProduct(LieAlgebraModule,LieAlgebraModule,ZZ) -- computes the multiplicities of irreducibles in the decomposition of the fusion product of U and V
• gtInvariantInVtensorVdual -- computes an invariant in $(V \otimes V^*)$ in the type A Gelfand-Tsetlin basis
• gtInvariantInVtensorVdual(List) -- computes an invariant in $(V \otimes V^*)$ in the type A Gelfand-Tsetlin basis
• GTPattern -- class for a Gelfand-Tsetlin pattern
• gtPatternFromEntries -- creates an object of type GTPattern from a list of entries
• gtPatternFromEntries(String,List) -- creates an object of type GTPattern from a list of entries
• gtPatterns -- a list of Gelfand-Tsetlin patterns of shape lambda
• gtPatterns(String,List) -- a list of Gelfand-Tsetlin patterns of shape lambda
• gtPolytope -- the polytope defined by the inequalities and equations appearing in the definition of Gelfand-Tsetlin patterns
• gtPolytope(String,List) -- the polytope defined by the inequalities and equations appearing in the definition of Gelfand-Tsetlin patterns
• GTrepresentationMatrices -- creates a list of matrices for the action of $\mathfrak{g}$ on Gelfand-Tsetlin basis
• GTrepresentationMatrices(LieAlgebraModule) -- creates a list of matrices for the action of $\mathfrak{g}$ on Gelfand-Tsetlin basis
• halfspinRepresentationMatrices -- matrix generators for the halfspin representations of $\mathfrak{so}(2n)$
• halfspinRepresentationMatrices(...,CoefficientRing=>...) -- matrix generators for the halfspin representations of $\mathfrak{so}(2n)$
• halfspinRepresentationMatrices(ZZ,ZZ) -- matrix generators for the halfspin representations of $\mathfrak{so}(2n)$
• highestRoot -- the highest root of a simple Lie algebra
• highestRoot(LieAlgebra) -- the highest root of a simple Lie algebra
• irreducibleLieAlgebraModule -- construct the irreducible Lie algebra module with given highest weight
• irreducibleLieAlgebraModule(LieAlgebra,List) -- construct the irreducible Lie algebra module with given highest weight
• irreducibleLieAlgebraModule(LieAlgebra,Vector) -- construct the irreducible Lie algebra module with given highest weight
• isIrreducible -- Whether a Lie algebra module is irreducible or not
• isIrreducible(LieAlgebraModule) -- Whether a Lie algebra module is irreducible or not
• isIsomorphic -- tests whether two Lie algebra are isomorphic
• isIsomorphic(LieAlgebra,LieAlgebra) -- tests whether two Lie algebra are isomorphic
• isLieAlgebraRepresentation -- checks whether a list of matrices defines a Lie algebra representation
• isLieAlgebraRepresentation(LieAlgebraBasis,List) -- checks whether a list of matrices defines a Lie algebra representation
• isomorphismOfRepresentations -- compute an explicit isomorphism between two Lie algebra representations
• isomorphismOfRepresentations(LieAlgebraRepresentation,LieAlgebraRepresentation) -- compute an explicit isomorphism between two Lie algebra representations
• killingForm -- computes the scaled Killing form applied to two weights
• killingForm(LieAlgebra,List,List) -- computes the scaled Killing form applied to two weights
• killingForm(LieAlgebra,Vector,Vector) -- computes the scaled Killing form applied to two weights
• LieAlgebra -- class for Lie algebras
• LieAlgebra ++ LieAlgebra -- Take the direct sum of Lie algebras
• LieAlgebra == LieAlgebra -- tests equality of LieAlgebra
• LieAlgebra _ ZZ -- selects one summand of a semi-simple Lie Algebra
• LieAlgebra _* -- gives the list of summands of a semi-simple Lie Algebra
• LieAlgebraBasis -- class for an enhanced Lie algebra basis
• lieAlgebraBasis -- computes an enhanced basis for a simple Lie algebra
• lieAlgebraBasis(LieAlgebra) -- computes an enhanced basis for a simple Lie algebra
• lieAlgebraBasis(String,ZZ) -- computes an enhanced basis for a simple Lie algebra
• LieAlgebraModule -- class for Lie algebra modules
• LieAlgebraModule ** LieAlgebraModule -- tensor product of LieAlgebraModules
• LieAlgebraModule ++ LieAlgebraModule -- direct sum of LieAlgebraModules
• LieAlgebraModule @ LieAlgebraModule -- Take the tensor product of modules over different Lie algebras
• LieAlgebraModule ^** ZZ -- Computes the nth tensor power of a Lie algebra module
• LieAlgebraModule _ LieAlgebraModule -- Pick out one irreducible submodule of a Lie algebra module
• LieAlgebraModule _ List -- Pick out one irreducible submodule of a Lie algebra module
• LieAlgebraModule _ Vector -- Pick out one irreducible submodule of a Lie algebra module
• LieAlgebraModule _ ZZ -- Pick out one irreducible submodule of a Lie algebra module
• LieAlgebraModule _* -- List irreducible submodules of a Lie algebra module
• LieAlgebraModuleFromWeights -- finds a Lie algebra module based on its weights
• LieAlgebraModuleFromWeights(RingElement,LieAlgebra) -- finds a Lie algebra module based on its weights
• LieAlgebraModuleFromWeights(VirtualTally,LieAlgebra) -- finds a Lie algebra module based on its weights
• LieAlgebraRepresentation -- class for a Lie algebra representation
• lieAlgebraRepresentation -- create a LieAlgebraRepresentation
• LieAlgebraRepresentation ** LieAlgebraRepresentation -- computes the explicit action on $V \otimes W$ given $\mathfrak{g}$-representations $V$ and $W$
• lieAlgebraRepresentation(LieAlgebraModule,LieAlgebraBasis,List) -- create a LieAlgebraRepresentation
• LieAlgebraRepresentations -- Lie algebra representations and characters
• LL -- construct the irreducible Lie algebra module with given highest weight
• multiplicity(...,BasisElementLimit=>...) -- compute the multiplicity of a weight in a Lie algebra module
• multiplicity(...,DegreeLimit=>...) -- compute the multiplicity of a weight in a Lie algebra module
• multiplicity(...,MinimalGenerators=>...) -- compute the multiplicity of a weight in a Lie algebra module
• multiplicity(...,PairLimit=>...) -- compute the multiplicity of a weight in a Lie algebra module
• multiplicity(...,Strategy=>...) -- compute the multiplicity of a weight in a Lie algebra module
• multiplicity(...,Variable=>...) -- compute the multiplicity of a weight in a Lie algebra module
• multiplicity(List,LieAlgebraModule) -- compute the multiplicity of a weight in a Lie algebra module
• multiplicity(Vector,LieAlgebraModule) -- compute the multiplicity of a weight in a Lie algebra module
• new LieAlgebra from Matrix -- Define a Lie algebra from its Cartan matrix
• positiveCoroots -- the positive (co)roots of a simple Lie algebra
• positiveCoroots(LieAlgebra) -- the positive (co)roots of a simple Lie algebra
• positiveRoots -- the positive (co)roots of a simple Lie algebra
• positiveRoots(LieAlgebra) -- the positive (co)roots of a simple Lie algebra
• qdim -- Compute principal specialization of character or quantum dimension
• qdim(LieAlgebraModule) -- Compute principal specialization of character or quantum dimension
• qdim(LieAlgebraModule,ZZ) -- Compute principal specialization of character or quantum dimension
• representationWeights -- computes the weights of the basis of a Lie algebra module from an explicit representation
• representationWeights(LieAlgebraRepresentation) -- computes the weights of the basis of a Lie algebra module from an explicit representation
• reynoldsOperator -- computes the projection to the sum of the trivial submodules in $V$
• reynoldsOperator(LieAlgebraRepresentation) -- computes the projection to the sum of the trivial submodules in $V$
• simpleLieAlgebra -- construct a simple Lie algebra
• simpleLieAlgebra(String,ZZ) -- construct a simple Lie algebra
• simpleRoots -- the simple roots of a simple Lie algebra
• simpleRoots(LieAlgebra) -- the simple roots of a simple Lie algebra
• simpleRoots(String,ZZ) -- the simple roots of a simple Lie algebra
• spinRepresentationMatrices -- matrix generators for the spin representation of $\mathfrak{so}(2n)$
• spinRepresentationMatrices(...,CoefficientRing=>...) -- matrix generators for the spin representation of $\mathfrak{so}(2n)$
• spinRepresentationMatrices(ZZ) -- matrix generators for the spin representation of $\mathfrak{so}(2n)$
• standardModule (missing documentation)
• standardRepresentation -- creates the standard representation of a matrix Lie algebra
• standardRepresentation(LieAlgebra) -- creates the standard representation of a matrix Lie algebra
• standardRepresentation(String,ZZ) -- creates the standard representation of a matrix Lie algebra
• starInvolution -- computes w* for a weight w
• starInvolution(LieAlgebraModule) -- computes w* for a weight w
• subLieAlgebra -- Define a sub-Lie algebra of an existing one
• subLieAlgebra(LieAlgebra,List) -- Define a sub-Lie algebra of an existing one
• subLieAlgebra(LieAlgebra,Matrix) -- Define a sub-Lie algebra of an existing one
• subLieAlgebra(LieAlgebra,String) -- Define a sub-Lie algebra of an existing one
• symmetricPower(ZZ,LieAlgebraModule) -- Computes the nth symmetric / exterior tensor power of a Lie algebra module
• symmetricPower(ZZ,LieAlgebraRepresentation) -- computes the explicit action on $\operatorname{Sym}^d V$ for a $\mathfrak{g}$-module $V$
• tensor(LieAlgebraRepresentation,LieAlgebraRepresentation) -- computes the explicit action on $V \otimes W$ given $\mathfrak{g}$-representations $V$ and $W$
• tensorCoefficient -- computes the multiplicity of W in U tensor V
• tensorCoefficient(LieAlgebraModule,LieAlgebraModule,LieAlgebraModule) -- computes the multiplicity of W in U tensor V
• trivialModule -- The trivial module of a Lie algebra
• trivialModule(LieAlgebra) -- The trivial module of a Lie algebra
• trivialRepresentation -- creates the trivial representation of a Lie algebra
• trivialRepresentation(LieAlgebra) -- creates the trivial representation of a Lie algebra
• trivialRepresentation(LieAlgebraBasis) -- creates the trivial representation of a Lie algebra
• trivialRepresentation(String,ZZ) -- creates the trivial representation of a Lie algebra
• UInVtensorW -- computes a basis of a submodule of $V \otimes W$ isomorphic to $U$ with a given highest weight vector
• UInVtensorW(LieAlgebraRepresentation,LieAlgebraRepresentation,LieAlgebraRepresentation,Matrix) -- computes a basis of a submodule of $V \otimes W$ isomorphic to $U$ with a given highest weight vector
• universalEnvelopingAlgebra -- computes the universal enveloping algebra of a Lie algebra
• universalEnvelopingAlgebra(LieAlgebra) -- computes the universal enveloping algebra of a Lie algebra
• universalEnvelopingAlgebra(LieAlgebraBasis) -- computes the universal enveloping algebra of a Lie algebra
• uNminus -- computes the universal enveloping algebra of the Lie algebra $N^{-}$
• uNminus(LieAlgebra) -- computes the universal enveloping algebra of the Lie algebra $N^{-}$
• uNminus(LieAlgebraBasis) -- computes the universal enveloping algebra of the Lie algebra $N^{-}$
• VInSymdW -- computes a basis of a submodule of $\operatorname{Sym}^d W$ isomorphic to $V$ with a given highest weight vector
• VInSymdW(LieAlgebraRepresentation,ZZ,LieAlgebraRepresentation,Matrix) -- computes a basis of a submodule of $\operatorname{Sym}^d W$ isomorphic to $V$ with a given highest weight vector
• VInWedgekW -- computes a basis of a submodule of $\bigwedge^k W$ isomorphic to $V$ with a given highest weight vector
• VInWedgekW(LieAlgebraRepresentation,ZZ,LieAlgebraRepresentation,Matrix) -- computes a basis of a submodule of $\bigwedge^k W$ isomorphic to $V$ with a given highest weight vector
• weightDiagram -- computes the weights in a Lie algebra module and their multiplicities
• weightDiagram(...,Strategy=>...) -- computes the weights in a Lie algebra module and their multiplicities
• weightDiagram(LieAlgebra,List) -- computes the weights in a Lie algebra module and their multiplicities
• weightDiagram(LieAlgebra,Vector) -- computes the weights in a Lie algebra module and their multiplicities
• weightDiagram(LieAlgebraModule) -- computes the weights in a Lie algebra module and their multiplicities
• weightMuHighestWeightVectorsInSymdW -- computes the highest weight vectors of weight mu in $\operatorname{Sym}^d W$
• weightMuHighestWeightVectorsInSymdW(List,ZZ,LieAlgebraRepresentation) -- computes the highest weight vectors of weight mu in $\operatorname{Sym}^d W$
• weightMuHighestWeightVectorsInW -- computes the highest weight vectors of weight mu in W
• weightMuHighestWeightVectorsInW(List,LieAlgebraRepresentation) -- computes the highest weight vectors of weight mu in W
• weightNuHighestWeightVectorsInVtensorW -- computes the highest weight vectors of weight nu in $V \otimes W$
• weightNuHighestWeightVectorsInVtensorW(List,LieAlgebraRepresentation,LieAlgebraRepresentation) -- computes the highest weight vectors of weight nu in $V \otimes W$
• weylAlcove -- the dominant integral weights of level less than or equal to l
• weylAlcove(LieAlgebra,ZZ) -- the dominant integral weights of level less than or equal to l
• weylAlcove(String,ZZ,ZZ) -- the dominant integral weights of level less than or equal to l
• weylAlcove(ZZ,LieAlgebra) -- the dominant integral weights of level less than or equal to l
• zeroModule -- The zero module of a Lie algebra
• zeroModule(LieAlgebra) -- The zero module of a Lie algebra