Symmetric Functions

• Introduction
  • Example: Symmetric Functions in Any Number of Variables
• Creation
  • Creation of Symmetric Function Algebras
    • SymmetricFunctionAlgebra(R): Rng → AlgSym
    • SFA(R): Rng → AlgSym
    • SymmetricFunctionAlgebraSchur(R): Rng → AlgSym
    • SFASchur(R): Rng → AlgSym
    • SymmetricFunctionAlgebraHomogeneous(R): Rng → AlgSym
    • SFAHomogeneous(R): Rng → AlgSym
    • SymmetricFunctionAlgebraPower(R): Rng → AlgSym
    • SFAPower(R): Rng → AlgSym
    • SymmetricFunctionAlgebraElementary(R): Rng → AlgSym
    • SFAElementary(R): Rng → AlgSym
    • SymmetricFunctionAlgebraMonomial(R): Rng → AlgSym
    • SFAMonomial(R): Rng → AlgSym
    • Example: Symmetric Function Algebras with Different Bases
  • Creation of Symmetric Functions
    • A . P: AlgSym, [ RngIntElt ] → AlgSymElt
    • A . i: AlgSym, RngIntElt → AlgSymElt
    • Example: Symmetric Functions as Linear Combinations
    • A ! f: AlgSym, RngMPolElt → AlgSymElt
    • Example: Coercing Symmetric Polynomials
    • Example: Redundant Terms after Coercion
    • A ! r: AlgSym, RngElt → AlgSymElt
    • Example: Scalar Symmetric Functions
    • A ! m: AlgSym, AlgSymElt → AlgSymElt
    • Example: Changing the Basis of a Symmetric Function
    • Example: Homogeneous Functions as Sums of Monomial Functions
• Structure Operations
  • Related Structures
    • BaseRing(L): AlgSym → Rng
    • CoefficientRing(L): AlgSym → Rng
    • Category(L): AlgSym → Cat
    • Parent(L): AlgSym → Pow
    • PrimeRing(L): AlgSym → Rng
  • Ring Predicates and Booleans
    • IsCommutative(L): AlgSym → BoolElt
    • IsUnitary(L): AlgSym → BoolElt
    • IsFinite(L): AlgSym → BoolElt
    • IsOrdered(L): AlgSym → BoolElt
    • IsField(L): AlgSym → BoolElt
    • IsEuclideanDomain(L): AlgSym → BoolElt
    • IsPID(L): AlgSym → BoolElt
    • IsUFD(L): AlgSym → BoolElt
    • IsDivisionRing(L): AlgSym → BoolElt
    • IsEuclideanRing(L): AlgSym → BoolElt
    • IsDomain(L): AlgSym → BoolElt
    • IsPrincipalIdealRing(L): AlgSym → BoolElt
    • L eq M: AlgSym, AlgSym → BoolElt
    • L ne M: AlgSym, AlgSym → BoolElt
  • Predicates on Basis Types
    • HasSchurBasis(A): AlgSym → BoolElt
    • HasHomogeneousBasis(A): AlgSym → BoolElt
    • HasElementaryBasis(A): AlgSym → BoolElt
    • HasPowerSumBasis(A): AlgSym → BoolElt
    • HasMonomialBasis(A): AlgSym → BoolElt
• Element Operations
  • Parent and Category
    • Parent(f): AlgSymElt → AlgSym
    • Category(f): AlgSymElt → Cat
  • Print Styles
    • A ` PrintStyle: AlgSym → MonStgElt
    • Example: Setting the Print Style
  • Additive Arithmetic Operators
    • + a: AlgSymElt → AlgSymElt
    • - a: AlgSymElt → AlgSymElt
    • a + b: AlgSymElt, AlgSymElt → AlgSymElt
    • a - b: AlgSymElt, AlgSymElt → AlgSymElt
    • a +:= b: AlgSymElt, AlgSymElt → AlgSymElt
    • a -:= b: AlgSymElt, AlgSymElt → AlgSymElt
  • Multiplication
    • a * b: AlgSymElt, AlgSymElt → AlgSymElt
    • a *:= b: AlgSymElt, AlgSymElt → AlgSymElt
    • a ^ k: AlgSymElt, RngIntElt → AlgSymElt
    • Example: Multiplication of Symmetric Functions
  • Plethysm
    • a ~ b: AlgSymElt, AlgSymElt → AlgSymElt
    • Example: Plethysm of Symmetric Functions
  • Boolean Operators
    • IsHomogeneous(s): AlgSymElt → BoolElt
    • IsZero(s): AlgSymElt → BoolElt
    • IsOne(s): AlgSymElt → BoolElt
    • IsMinusOne(s): AlgSymElt → BoolElt
    • s eq t: AlgSymElt, AlgSymElt → BoolElt
    • s ne t: AlgSymElt, AlgSymElt → BoolElt
  • Accessing Elements
    • Coefficient(s, p): AlgSymElt, SeqEnum → RngElt
    • Support(s): AlgSymElt → [SeqEnum], [RngElt]
    • Length(s): AlgSymElt → RngIntElt
    • Example: Decomposing and Rebuilding an Element
    • Degree(s): AlgSymElt → RngIntElt
  • Multivariate Polynomials
    • P ! s: RngMPol, AlgSymElt → RngMPolElt
    • Example: Restricting to Finitely Many Variables
    • Example: Change of Alphabet
  • Frobenius Homomorphism
    • Frobenius(s): AlgSymElt → AlgSymElt
    • Example: Frobenius Automorphism
  • Inner Product
    • InnerProduct(a,b): AlgSymElt, AlgSymElt → RngElt
    • Example: Inner Product of Symmetric Functions
  • Combinatorial Objects
    • Tableaux(sf, m): AlgSymElt, RngIntElt → SetMulti
  • Symmetric Group Character
    • SymmetricCharacter(sf): AlgSymElt → AlgChtrElt
    • Example: Characters Induced from Young Subgroups
  • 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: Kostka Numbers
    • SchurToHomogeneousMatrix(n): RngIntElt → AlgMatElt
    • SchurToPowerSumMatrix(n): RngIntElt → AlgMatElt
    • SchurToElementaryMatrix(n): RngIntElt → AlgMatElt
    • Example: Transition Matrices Agree with Coercion
  • 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: Homogeneous-to-Schur Matrix as a Transpose
    • HomogeneousToMonomialMatrix(n): RngIntElt → AlgMatElt
    • Example: Symmetry of the Homogeneous-to-Monomial Matrix
    • HomogeneousToPowerSumMatrix(n): RngIntElt → AlgMatElt
    • HomogeneousToElementaryMatrix(n): RngIntElt → AlgMatElt
    • Example: Homogeneous-to-Elementary Transition Matrix
  • Transition Matrices from Power Sum Basis
    • PowerSumToSchurMatrix(n): RngIntElt → AlgMatElt
    • Example: Character Table of the Symmetric Group
    • 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: Symmetry of the Elementary-to-Monomial Matrix
    • ElementaryToHomogeneousMatrix(n): RngIntElt → AlgMatElt
    • ElementaryToPowerSumMatrix(n): RngIntElt → AlgMatElt
• Bibliography