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Packages ยป LieAlgebraRepresentations :: LieAlgebraRepresentations
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LieAlgebraRepresentations -- Lie algebra representations and characters

Description

This package implements finite-dimensional representations of finite-dimensional complex semisimple Lie algebras and their characters.

This package is a major expansion and renaming of the package formerly known as LieTypes.

Authors

Certification a gold star

Version 0.5 of this package (under the name "LieTypes") was accepted for publication in volume 8 of The Journal of Software for Algebra and Geometry on 2 August 2018, in the article Software for computing conformal block divisors on bar M_0,n (DOI: 10.2140/jsag.2018.8.81). That version can be obtained from the journal.

Version

This documentation describes version 1.0 of LieAlgebraRepresentations, released Nov 8, 2025.

Citation

If you have used this package in your research, please cite it as follows:

@misc{LieAlgebraRepresentationsSource,
  title = {{LieAlgebraRepresentations: Lie algebra representations and characters. Version~1.0}},
  author = {Dave Swinarski and Paul Zinn-Justin},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

@article{LieAlgebraRepresentationsArticle,
  title = {{Software for computing conformal block divisors on bar M_0,n}},
  author = {Dave Swinarski and Paul Zinn-Justin},
  journal = {The Journal of Software for Algebra and Geometry},
  volume = {8},
  year = {2018},
}

Exports

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

For the programmer

The object LieAlgebraRepresentations is a package, defined in LieAlgebraRepresentations.m2, with auxiliary files in LieAlgebraRepresentations/.

The source of this document is in LieAlgebraRepresentations/documentation.m2:12:0.