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SYMMETRIC FUNCTIONS

 
Acknowledgements
 
Introduction
 
Creation
      Creation of Symmetric Function Algebras
      Creation of Symmetric Functions
 
Structure Operations
      Related Structures
      Ring Predicates and Booleans
      Predicates on Basis Types
 
Element Operations
      Parent and Category
      Print Styles
      Additive Arithmetic Operators
      Multiplication
      Plethysm
      Boolean Operators
      Accessing Elements
      Multivariate Polynomials
      Frobenius Homomorphism
      Inner Product
      Combinatorial Objects
      Symmetric Group Character
      Restrictions
 
Transition Matrices
      Transition Matrices from Schur Basis
      Transition Matrices from Monomial Basis
      Transition Matrices from Homogeneous Basis
      Transition Matrices from Power Sum Basis
      Transition Matrices from Elementary Basis
 
Bibliography







DETAILS

 
Introduction
      Example AlgSym_inf-invar (H146E1)

 
Creation

      Creation of Symmetric Function Algebras
            SymmetricFunctionAlgebra(R) : Rng -> AlgSym
            SymmetricFunctionAlgebraSchur(R) : Rng -> AlgSym
            SymmetricFunctionAlgebraHomogeneous(R) : Rng -> AlgSym
            SymmetricFunctionAlgebraPower(R) : Rng -> AlgSym
            SymmetricFunctionAlgebraElementary(R) : Rng -> AlgSym
            SymmetricFunctionAlgebraMonomial(R) : Rng -> AlgSym
            Example AlgSym_creation (H146E2)

      Creation of Symmetric Functions
            A . P : AlgSym, [ RngIntElt ] -> AlgSymElt
            A . i : AlgSym, RngIntElt -> AlgSymElt
            Example AlgSym_elt-create (H146E3)
            IsCoercible(A, f) : AlgSym, RngMPolElt -> BoolElt, AlgSymElt
            Example AlgSym_poly bang (H146E4)
            A ! r : AlgSym, RngElt -> AlgSymElt
            Example AlgSym_elt-create-scalar (H146E5)
            A ! m : AlgSym, AlgSymElt -> AlgSymElt
            Example AlgSym_elt-create-change_basis (H146E6)
            Example AlgSym_change-basis-2 (H146E7)

 
Structure Operations

      Related Structures
            BaseRing(L) : AlgSym -> Rng

      Ring Predicates and Booleans
            L eq M : AlgSym, AlgSym -> BoolElt

      Predicates on Basis Types
            HasSchurBasis(A): AlgSym -> BoolElt
            HasHomogeneousBasis(A): AlgSym -> BoolElt

 
Element Operations

      Parent and Category

      Print Styles
            A ` PrintStyle : AlgSym -> MonStgElt
            Example AlgSym_print-style (H146E8)

      Additive Arithmetic Operators

      Multiplication
            a * b : AlgSymElt, AlgSymElt -> AlgSymElt
            Example AlgSym_mult (H146E9)

      Plethysm
            a ~ b : AlgSymElt, AlgSymElt -> AlgSymElt
            Example AlgSym_pleth (H146E10)

      Boolean Operators
            IsHomogeneous(s): AlgSymElt -> BoolElt
            s eq t : AlgSymElt, AlgSymElt -> BoolElt

      Accessing Elements
            Coefficient(s, p) : AlgSymElt, SeqEnum -> RngElt
            Support(s) : AlgSymElt -> [SeqEnum], [RngElt]
            Length(s) : AlgSymElt -> RngIntElt
            Example AlgSym_support (H146E11)
            Degree(s) : AlgSymElt -> RngIntElt

      Multivariate Polynomials
            P ! s : RngMPol, AlgSymElt -> RngMPolElt
            Example AlgSym_symmetric polynomials and symmetric functions (H146E12)
            Example AlgSym_ChangeOfAlphabet (H146E13)

      Frobenius Homomorphism
            Frobenius(s) : AlgSymElt -> AlgSymElt
            Example AlgSym_Frobenius automorphism (H146E14)

      Inner Product
            InnerProduct(a,b): AlgSymElt, AlgSymElt -> RngElt
            Example AlgSym_Inner-Product (H146E15)

      Combinatorial Objects
            Tableaux(sf, m): AlgSymElt, RngIntElt -> SetMulti

      Symmetric Group Character
            SymmetricCharacter(sf): AlgSymElt -> AlgChtrElt
            Example AlgSym_symmetric-polynomials-and-symmetric-functions (H146E16)

      Restrictions
            RestrictDegree(a, n): AlgSymElt, RngIntElt -> AlgSymElt
            RestrictPartitionLength(a, n): AlgSymElt, RngIntElt -> AlgSymElt
            RestrictParts(a, n): AlgSymElt, RngIntElt -> AlgSymElt

 
Transition Matrices

      Transition Matrices from Schur Basis
            SchurToMonomialMatrix(n): RngIntElt -> AlgMatElt
            Example AlgSym_creation (H146E17)
            SchurToHomogeneousMatrix(n): RngIntElt -> AlgMatElt
            SchurToPowerSumMatrix(n): RngIntElt -> AlgMatElt
            SchurToElementaryMatrix(n): RngIntElt -> AlgMatElt
            Example AlgSym_trans (H146E18)

      Transition Matrices from Monomial Basis
            MonomialToSchurMatrix(n): RngIntElt -> AlgMatElt
            MonomialToHomogeneousMatrix(n): RngIntElt -> AlgMatElt
            MonomialToPowerSumMatrix(n): RngIntElt -> AlgMatElt
            MonomialToElementaryMatrix(n): RngIntElt -> AlgMatElt

      Transition Matrices from Homogeneous Basis
            HomogeneousToSchurMatrix(n): RngIntElt -> AlgMatElt
            Example AlgSym_HS (H146E19)
            HomogeneousToMonomialMatrix(n): RngIntElt -> AlgMatElt
            Example AlgSym_hm (H146E20)
            HomogeneousToPowerSumMatrix(n): RngIntElt -> AlgMatElt
            HomogeneousToElementaryMatrix(n): RngIntElt -> AlgMatElt
            Example AlgSym_HE (H146E21)

      Transition Matrices from Power Sum Basis
            PowerSumToSchurMatrix(n): RngIntElt -> AlgMatElt
            Example AlgSym_character-table (H146E22)
            PowerSumToMonomialMatrix(n): RngIntElt -> AlgMatElt
            PowerSumToHomogeneousMatrix(n): RngIntElt -> AlgMatElt
            PowerSumToElementaryMatrix(n): RngIntElt -> AlgMatElt

      Transition Matrices from Elementary Basis
            ElementaryToSchurMatrix(n): RngIntElt -> AlgMatElt
            ElementaryToMonomialMatrix(n): RngIntElt -> AlgMatElt
            Example AlgSym_em (H146E23)
            ElementaryToHomogeneousMatrix(n): RngIntElt -> AlgMatElt
            ElementaryToPowerSumMatrix(n): RngIntElt -> AlgMatElt

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013