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# LieAlgebraRepresentations : Index

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

- 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