Introduction
      Example AlgSym_inf-invar (H159E1)

 
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 (H159E2)

      Creation of Symmetric Functions
            A . P : AlgSym, [ RngIntElt ] -> AlgSymElt
            A . i : AlgSym, RngIntElt -> AlgSymElt
            Example AlgSym_elt-create (H159E3)
            A ! f : AlgSym, RngMPolElt -> AlgSymElt
            Example AlgSym_poly bang (H159E4)
            Example AlgSym_poly-bang-2 (H159E5)
            A ! r : AlgSym, RngElt -> AlgSymElt
            Example AlgSym_elt-create-scalar (H159E6)
            A ! m : AlgSym, AlgSymElt -> AlgSymElt
            Example AlgSym_elt-create-change_basis (H159E7)
            Example AlgSym_change-basis-2 (H159E8)

 
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 (H159E9)

      Additive Arithmetic Operators

      Multiplication
            a * b : AlgSymElt, AlgSymElt -> AlgSymElt
            Example AlgSym_mult (H159E10)

      Plethysm
            a ~ b : AlgSymElt, AlgSymElt -> AlgSymElt
            Example AlgSym_pleth (H159E11)

      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 (H159E12)
            Degree(s) : AlgSymElt -> RngIntElt

      Multivariate Polynomials
            P ! s : RngMPol, AlgSymElt -> RngMPolElt
            Example AlgSym_symmetric polynomials and symmetric functions (H159E13)
            Example AlgSym_ChangeOfAlphabet (H159E14)

      Frobenius Homomorphism
            Frobenius(s) : AlgSymElt -> AlgSymElt
            Example AlgSym_Frobenius automorphism (H159E15)

      Inner Product
            InnerProduct(a,b): AlgSymElt, AlgSymElt -> RngElt
            Example AlgSym_Inner-Product (H159E16)

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

      Symmetric Group Character
            SymmetricCharacter(sf): AlgSymElt -> AlgChtrElt
            Example AlgSym_symmetric-polynomials-and-symmetric-functions (H159E17)

      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 (H159E18)
            SchurToHomogeneousMatrix(n): RngIntElt -> AlgMatElt
            SchurToPowerSumMatrix(n): RngIntElt -> AlgMatElt
            SchurToElementaryMatrix(n): RngIntElt -> AlgMatElt
            Example AlgSym_trans (H159E19)

      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 (H159E20)
            HomogeneousToMonomialMatrix(n): RngIntElt -> AlgMatElt
            Example AlgSym_hm (H159E21)
            HomogeneousToPowerSumMatrix(n): RngIntElt -> AlgMatElt
            HomogeneousToElementaryMatrix(n): RngIntElt -> AlgMatElt
            Example AlgSym_HE (H159E22)

      Transition Matrices from Power Sum Basis
            PowerSumToSchurMatrix(n): RngIntElt -> AlgMatElt
            Example AlgSym_character-table (H159E23)
            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 (H159E24)
            ElementaryToHomogeneousMatrix(n): RngIntElt -> AlgMatElt
            ElementaryToPowerSumMatrix(n): RngIntElt -> AlgMatElt

 
Bibliography

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