LieAlgebraRepresentations : Table of Contents

• LieAlgebraRepresentations -- Lie algebra representations and characters
• 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
• adjointModule -- The adjoint module of a Lie algebra
• adjointRepresentation -- 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
• branchingRule -- A Lie algebra module viewed as a module over a Lie subalgebra
• cartanMatrix -- Provide the Cartan matrix of a simple Lie algebra
• casimirOperator -- computes the Casimir operator associated to a representation
• casimirProjection -- 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
• casimirSpectrum -- computes the eigenvalues of the Casimir operator associated to a representation
• character -- Computes the character of a Lie algebra module
• characterRing (missing documentation)
• deGraafBases -- compute the bases produced by de Graaf's algorithm
• deGraafRepresentation -- 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
• dualCoxeterNumber -- the dual Coxeter number of a simple Lie algebra
• dynkinDiagram -- 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
• embedding -- gives the embedding of Cartan subalgebras of one Lie algebra into another
• 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
• fusionProduct -- 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
• GTPattern -- class for a Gelfand-Tsetlin pattern
• gtPatternFromEntries -- creates an object of type GTPattern from a list of entries
• gtPatterns -- 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
• GTrepresentationMatrices -- creates a list of matrices for the action of $\mathfrak{g}$ on Gelfand-Tsetlin basis
• halfspinRepresentationMatrices -- matrix generators for the halfspin representations of $\mathfrak{so}(2n)$
• highestRoot -- the highest root of a simple Lie algebra
• irreducibleLieAlgebraModule -- construct the irreducible Lie algebra module with given highest weight
• isIrreducible -- Whether a Lie algebra module is irreducible or not
• isIsomorphic -- tests whether two Lie algebra are isomorphic
• isLieAlgebraRepresentation -- checks whether a list of matrices defines a Lie algebra representation
• isomorphismOfRepresentations -- compute an explicit isomorphism between two Lie algebra representations
• killingForm -- 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
• 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 _ 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
• 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 $W$
• multiplicity(List,LieAlgebraModule) -- compute the multiplicity of a weight in a Lie algebra module
• new LieAlgebra from Matrix -- Define a Lie algebra from its Cartan matrix
• positiveRoots -- the positive (co)roots of a simple Lie algebra
• qdim -- Compute principal specialization of character or quantum dimension
• representationWeights -- 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 $V$
• simpleLieAlgebra -- construct a simple Lie algebra
• simpleRoots -- the simple roots of a simple Lie algebra
• spinRepresentationMatrices -- matrix generators for the spin representation of $\mathfrak{so}(2n)$
• standardModule (missing documentation)
• standardRepresentation -- creates the standard representation of a matrix Lie algebra
• starInvolution -- computes w* for a weight w
• subLieAlgebra -- 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 $V$
• tensorCoefficient -- computes the multiplicity of W in U tensor V
• trivialModule -- The trivial module of a Lie algebra
• trivialRepresentation -- 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
• universalEnvelopingAlgebra -- computes the universal enveloping algebra of a Lie algebra
• uNminus -- 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
• VInWedgekW -- 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
• weightMuHighestWeightVectorsInSymdW -- computes the highest weight vectors of weight mu in $\operatorname{Sym}^d W$
• weightMuHighestWeightVectorsInW -- computes the highest weight vectors of weight mu in W
• weightNuHighestWeightVectorsInVtensorW -- 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
• zeroModule -- The zero module of a Lie algebra