SpechtModule -- a package for constructing Specht Modules

Description

SpechtModule calculates many objects related to the irreducible representations of the symmetric functions. This construction is used to implement an algorithm in invariant theory which calculates efficiently the secondary invariants of any permutation group.

The main features of the package include a method for calculating the character table of $S_n$, algorithms for calculating list of tableaux given a partition (tabloids, standard tableaux and semistandard tableaux among others) an implementation of the straightening algorithm which includes an implementation of the Garnir element given a tableau an a row descent. Methods for calculating Higher Specht Polynomials which give a basis of the Specht Modules that arise in the coinvariant ring of $S_n$ which is the quotient $k[x_1,..,x_n]/({\rm Sym}(n)^+)$. And finally methods for calculating the secondary invariants described above.

Caveat

An improvement can be made by finding an efficient way to calculate or represent Schur Polynomials

Author

Jonathan Niño <[email protected]>

Version

This documentation describes version 1.0 of SpechtModule.

Source code

The source code from which this documentation is derived is in the file SpechtModule.m2.

Exports

Types
- CharacterTable -- the class of character tables
- SpechtModuleElement -- the class of Specht Module elements
- TableauList -- the class of list of tableaux
- YoungTableau -- the class of Young Tableaux
Functions and commands
- addTableau -- see addTableau(TableauList,YoungTableau) -- adds a Young Tableau in the list
- cardinalityOfConjugacyClass -- see cardinalityOfConjugacyClass(Partition) -- the size of the conjugacy classes of S_n
- characterTable -- returns the character table of the symmetric group
- columnStabilizer -- see columnStabilizer(YoungTableau) -- the column stabilizer of the tableau
- conjugacyClass -- see conjugacyClass(List) -- the conjugacy class of a permutation
- cycleDecomposition -- see cycleDecomposition(List) -- the cycle decomposition of a permutation
- elementarySymmetricPolynomials -- see elementarySymmetricPolynomials(PolynomialRing) -- the elementary symmetric polynomials
- firstRowDescent -- see firstRowDescent(YoungTableau) -- retrieves the first row descent of a Young tableau
- garnirElement -- a SpechtModuleElement that is equal to zero
- generalizedVandermondeMatrix -- see generalizedVandermondeMatrix(List,List,PolynomialRing) -- the method for calculating generalized Vandermonde matrices
- generatePermutationGroup -- see generatePermutationGroup(List) -- a method for generating a permutation group given a set of generators
- higherSpechtPolynomial -- see higherSpechtPolynomial(YoungTableau,YoungTableau,PolynomialRing) -- the higher Specht polynomial index by the pair of standard tableaux (S,T)
- higherSpechtPolynomials -- a method that gives sets of higher Specht polynomials
- hookLengthFormula -- see hookLengthFormula(Partition) -- a formula for the number of standard tableaux
- indexMonomial -- see indexMonomial(YoungTableau,YoungTableau,PolynomialRing) -- a monomial that represents an index tableau
- indexTableau -- see indexTableau(YoungTableau) -- the index tableau for a given tableau
- innerProduct -- see innerProduct(ZZ,MutableMatrix,MutableMatrix) -- calculates the inner product for the characters of S_n
- listToTableau -- see listToTableau(List) -- constructs a Young Tableau from a doubly nested list of numbers
- matrixRepresentation -- the matrix representation of a permutation in the Specht Module
- multinomial -- a formula for the multinomial coefficient
- permutationMatrix -- see permutationMatrix(List) -- a permutation matrix generator
- permutationSign -- the sign of a permutation
- permutePolynomial -- permutes a RingElement or a PolynomialExpression of RingElements
- powerSumSymmetricPolynomials -- see powerSumSymmetricPolynomials(PolynomialRing) -- the power sum symmetric polynomials
- readingWord -- see readingWord(YoungTableau) -- gives the reading word of a given tableau
- representationMultiplicity -- the number of secondary invariants in a given irreducible representation
- rowPermutationTableaux -- see rowPermutationTableaux(YoungTableau) -- the list of row permutations without repetitions in columns
- rowStabilizer -- see rowStabilizer(YoungTableau) -- the row stabilizer of the tableau
- schurPolynomial -- see schurPolynomial(List,Partition,PolynomialRing) -- a method for constructing Schur polynomials
- secondaryInvariants -- see secondaryInvariants(List,PolynomialRing) -- the set of secondaryInvariants of a permutation group
- semistandardTableaux -- see semistandardTableaux(Partition,ZZ) -- the list of semistandard tableaux of shape p and filling with the numbers from 0 to n-1.
- sortColumnsTableau -- see sortColumnsTableau(SpechtModuleElement) -- a method for sorting the columns of the tableaux in a SpechtModuleElement
- spechtModuleElement -- the constructor for the class SpechtModuleElement
- spechtPolynomial -- see spechtPolynomial(YoungTableau,PolynomialRing) -- the Specht polynomial indexed by a standard tableau
- spechtPolynomials -- see spechtPolynomials(Partition,PolynomialRing) -- the set of all Specht polynomial indexed by standard tableaux of shape p
- standardTableaux -- see standardTableaux(Partition) -- the list of standard tableaux of shape p
- straighteningAlgorithm -- an algorithm for expressing any polytabloid as linear combinations of standard polytabloids
- tableauList -- see tableauList(Partition,ZZ) -- the constructor for the type TableauList
- tableauToList -- see tableauToList(YoungTableau) -- converts a YoungTableau to list form
- tabloids -- see tabloids(Partition) -- the list of tabloids for a given partition
- toListOfTableaux -- see toListOfTableaux(TableauList) -- converts an object of type TableauList into a list of YoungTableau objects
- vandermondeDeterminant -- the vandermonde determinant for a set of generators of a ring
- youngTableau -- the constructor method for the class YoungTableau
Methods
- addTableau(TableauList,List) -- see addTableau(TableauList,YoungTableau) -- adds a Young Tableau in the list
- addTableau(TableauList,YoungTableau) -- adds a Young Tableau in the list
- cardinalityOfConjugacyClass(Partition) -- the size of the conjugacy classes of S_n
- net(CharacterTable) -- see CharacterTable -- the class of character tables
- characterTable(ZZ) -- see characterTable -- returns the character table of the symmetric group
- CharacterTable _ Sequence -- retrieves an entry from the character table
- CharacterTable _ Sequence = Thing -- modifies an entry from the character table
- columnStabilizer(YoungTableau) -- the column stabilizer of the tableau
- conjugacyClass(List) -- the conjugacy class of a permutation
- cycleDecomposition(List) -- the cycle decomposition of a permutation
- elementarySymmetricPolynomials(PolynomialRing) -- the elementary symmetric polynomials
- entries(YoungTableau) -- returns the filling of the tableau
- firstRowDescent(YoungTableau) -- retrieves the first row descent of a Young tableau
- garnirElement(YoungTableau) -- see garnirElement -- a SpechtModuleElement that is equal to zero
- garnirElement(YoungTableau,ZZ) -- see garnirElement -- a SpechtModuleElement that is equal to zero
- garnirElement(YoungTableau,ZZ,ZZ,ZZ) -- see garnirElement -- a SpechtModuleElement that is equal to zero
- generalizedVandermondeMatrix(List,List,PolynomialRing) -- the method for calculating generalized Vandermonde matrices
- generatePermutationGroup(List) -- a method for generating a permutation group given a set of generators
- higherSpechtPolynomial(YoungTableau,YoungTableau,PolynomialRing) -- the higher Specht polynomial index by the pair of standard tableaux (S,T)
- higherSpechtPolynomials(Partition,PolynomialRing) -- see higherSpechtPolynomials -- a method that gives sets of higher Specht polynomials
- higherSpechtPolynomials(PolynomialRing) -- see higherSpechtPolynomials -- a method that gives sets of higher Specht polynomials
- higherSpechtPolynomials(YoungTableau,PolynomialRing) -- see higherSpechtPolynomials -- a method that gives sets of higher Specht polynomials
- higherSpechtPolynomials(YoungTableau,TableauList,PolynomialRing) -- see higherSpechtPolynomials -- a method that gives sets of higher Specht polynomials
- hookLengthFormula(Partition) -- a formula for the number of standard tableaux
- indexMonomial(YoungTableau,YoungTableau,PolynomialRing) -- a monomial that represents an index tableau
- indexTableau(YoungTableau) -- the index tableau for a given tableau
- innerProduct(ZZ,MutableMatrix,MutableMatrix) -- calculates the inner product for the characters of S_n
- listToTableau(List) -- constructs a Young Tableau from a doubly nested list of numbers
- matrixRepresentation(List,Partition) -- see matrixRepresentation -- the matrix representation of a permutation in the Specht Module
- matrixRepresentation(List,TableauList) -- see matrixRepresentation -- the matrix representation of a permutation in the Specht Module
- matrixRepresentation(Partition) -- see matrixRepresentation -- the matrix representation of a permutation in the Specht Module
- matrixRepresentation(TableauList) -- see matrixRepresentation -- the matrix representation of a permutation in the Specht Module
- multinomial(List) -- see multinomial -- a formula for the multinomial coefficient
- multinomial(Partition) -- see multinomial -- a formula for the multinomial coefficient
- multinomial(Tally) -- see multinomial -- a formula for the multinomial coefficient
- numColumns(YoungTableau) -- returns the number of columns of a tableau
- numRows(YoungTableau) -- returns the number of rows of a tableau
- permutationMatrix(List) -- a permutation matrix generator
- permutationSign(List) -- see permutationSign -- the sign of a permutation
- permutationSign(Partition) -- see permutationSign -- the sign of a permutation
- permutePolynomial(List,Minus) -- see permutePolynomial -- permutes a RingElement or a PolynomialExpression of RingElements
- permutePolynomial(List,Power) -- see permutePolynomial -- permutes a RingElement or a PolynomialExpression of RingElements
- permutePolynomial(List,Product) -- see permutePolynomial -- permutes a RingElement or a PolynomialExpression of RingElements
- permutePolynomial(List,RingElement) -- see permutePolynomial -- permutes a RingElement or a PolynomialExpression of RingElements
- permutePolynomial(List,Sum) -- see permutePolynomial -- permutes a RingElement or a PolynomialExpression of RingElements
- powerSumSymmetricPolynomials(PolynomialRing) -- the power sum symmetric polynomials
- readingWord(YoungTableau) -- gives the reading word of a given tableau
- representationMultiplicity(Tally,Partition) -- see representationMultiplicity -- the number of secondary invariants in a given irreducible representation
- representationMultiplicity(Tally,Partition,CharacterTable) -- see representationMultiplicity -- the number of secondary invariants in a given irreducible representation
- rowPermutationTableaux(YoungTableau) -- the list of row permutations without repetitions in columns
- rowStabilizer(YoungTableau) -- the row stabilizer of the tableau
- schurPolynomial(List,Partition,PolynomialRing) -- a method for constructing Schur polynomials
- secondaryInvariants(List,PolynomialRing) -- the set of secondaryInvariants of a permutation group
- semistandardTableaux(Partition,ZZ) -- the list of semistandard tableaux of shape p and filling with the numbers from 0 to n-1.
- size(YoungTableau) -- returns the number of cells of a tableau
- sortColumnsTableau(SpechtModuleElement) -- a method for sorting the columns of the tableaux in a SpechtModuleElement
- sortColumnsTableau(YoungTableau) -- a method for
- List SpechtModuleElement -- see SpechtModuleElement -- the class of Specht Module elements
- net(SpechtModuleElement) -- see SpechtModuleElement -- the class of Specht Module elements
- QQ * SpechtModuleElement -- see SpechtModuleElement -- the class of Specht Module elements
- SpechtModuleElement + SpechtModuleElement -- see SpechtModuleElement -- the class of Specht Module elements
- SpechtModuleElement - SpechtModuleElement -- see SpechtModuleElement -- the class of Specht Module elements
- terms(SpechtModuleElement) -- see SpechtModuleElement -- the class of Specht Module elements
- trim(SpechtModuleElement) -- see SpechtModuleElement -- the class of Specht Module elements
- ZZ * SpechtModuleElement -- see SpechtModuleElement -- the class of Specht Module elements
- spechtModuleElement(Partition,MutableHashTable) -- see spechtModuleElement -- the constructor for the class SpechtModuleElement
- spechtModuleElement(YoungTableau) -- see spechtModuleElement -- the constructor for the class SpechtModuleElement
- spechtModuleElement(YoungTableau,QQ) -- see spechtModuleElement -- the constructor for the class SpechtModuleElement
- spechtModuleElement(YoungTableau,ZZ) -- see spechtModuleElement -- the constructor for the class SpechtModuleElement
- spechtPolynomial(YoungTableau,PolynomialRing) -- the Specht polynomial indexed by a standard tableau
- spechtPolynomials(Partition,PolynomialRing) -- the set of all Specht polynomial indexed by standard tableaux of shape p
- standardTableaux(Partition) -- the list of standard tableaux of shape p
- straighteningAlgorithm(SpechtModuleElement) -- see straighteningAlgorithm -- an algorithm for expressing any polytabloid as linear combinations of standard polytabloids
- straighteningAlgorithm(YoungTableau) -- see straighteningAlgorithm -- an algorithm for expressing any polytabloid as linear combinations of standard polytabloids
- straighteningAlgorithm(YoungTableau,ZZ) -- see straighteningAlgorithm -- an algorithm for expressing any polytabloid as linear combinations of standard polytabloids
- net(TableauList) -- see TableauList -- the class of list of tableaux
- TableauList _ ZZ -- a method that retrieves the tableaux from the list
- tableauList(Partition) -- see tableauList(Partition,ZZ) -- the constructor for the type TableauList
- tableauList(Partition,ZZ) -- the constructor for the type TableauList
- tableauToList(YoungTableau) -- converts a YoungTableau to list form
- tabloids(Partition) -- the list of tabloids for a given partition
- toListOfTableaux(TableauList) -- converts an object of type TableauList into a list of YoungTableau objects
- vandermondeDeterminant(List,PolynomialRing) -- see vandermondeDeterminant -- the vandermonde determinant for a set of generators of a ring
- net(YoungTableau) -- see YoungTableau -- the class of Young Tableaux
- youngTableau(Partition) -- see youngTableau -- the constructor method for the class YoungTableau
- youngTableau(Partition,List) -- see youngTableau -- the constructor method for the class YoungTableau
- youngTableau(Partition,MutableList) -- see youngTableau -- the constructor method for the class YoungTableau
- YoungTableau == YoungTableau -- checks whether two tableaux are equivalent
- YoungTableau ? YoungTableau -- an order of YoungTableaux
- YoungTableau ^ ZZ -- retrieves a row from a Young Tableau
- YoungTableau _ Sequence -- retrieves the entry in cell (a,b) from a Young Tableau
- YoungTableau _ Sequence = Thing -- changes the entry in cell (a,b) from a Young Tableau
- YoungTableau _ ZZ -- retrieves a column from a Young Tableau
- youngTableau(YoungTableau) -- creates a copy of a YoungTableau object
Symbols
- AsExpression -- an optional argument that returns polynomials as expressions
- Robust -- an optional argument for specifying the algorithm for calculating higherSpechtPolynomials

For the programmer

The object SpechtModule is a package.