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Computer • algebra
Documentation
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Introduction
Example
AlgSym_inf-invar (H155E1)
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 (H155E2)
Creation of Symmetric Functions
A . P : AlgSym, [ RngIntElt ] -> AlgSymElt
A . i : AlgSym, RngIntElt -> AlgSymElt
Example
AlgSym_elt-create (H155E3)
A ! f : AlgSym, RngMPolElt -> AlgSymElt
Example
AlgSym_poly bang (H155E4)
Example
AlgSym_poly-bang-2 (H155E5)
A ! r : AlgSym, RngElt -> AlgSymElt
Example
AlgSym_elt-create-scalar (H155E6)
A ! m : AlgSym, AlgSymElt -> AlgSymElt
Example
AlgSym_elt-create-change_basis (H155E7)
Example
AlgSym_change-basis-2 (H155E8)
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 (H155E9)
Additive Arithmetic Operators
Multiplication
a * b : AlgSymElt, AlgSymElt -> AlgSymElt
Example
AlgSym_mult (H155E10)
Plethysm
a ~ b : AlgSymElt, AlgSymElt -> AlgSymElt
Example
AlgSym_pleth (H155E11)
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 (H155E12)
Degree(s) : AlgSymElt -> RngIntElt
Multivariate Polynomials
P ! s : RngMPol, AlgSymElt -> RngMPolElt
Example
AlgSym_symmetric polynomials and symmetric functions (H155E13)
Example
AlgSym_ChangeOfAlphabet (H155E14)
Frobenius Homomorphism
Frobenius(s) : AlgSymElt -> AlgSymElt
Example
AlgSym_Frobenius automorphism (H155E15)
Inner Product
InnerProduct(a,b): AlgSymElt, AlgSymElt -> RngElt
Example
AlgSym_Inner-Product (H155E16)
Combinatorial Objects
Tableaux(sf, m): AlgSymElt, RngIntElt -> SetMulti
Symmetric Group Character
SymmetricCharacter(sf): AlgSymElt -> AlgChtrElt
Example
AlgSym_symmetric-polynomials-and-symmetric-functions (H155E17)
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 (H155E18)
SchurToHomogeneousMatrix(n): RngIntElt -> AlgMatElt
SchurToPowerSumMatrix(n): RngIntElt -> AlgMatElt
SchurToElementaryMatrix(n): RngIntElt -> AlgMatElt
Example
AlgSym_trans (H155E19)
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 (H155E20)
HomogeneousToMonomialMatrix(n): RngIntElt -> AlgMatElt
Example
AlgSym_hm (H155E21)
HomogeneousToPowerSumMatrix(n): RngIntElt -> AlgMatElt
HomogeneousToElementaryMatrix(n): RngIntElt -> AlgMatElt
Example
AlgSym_HE (H155E22)
Transition Matrices from Power Sum Basis
PowerSumToSchurMatrix(n): RngIntElt -> AlgMatElt
Example
AlgSym_character-table (H155E23)
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 (H155E24)
ElementaryToHomogeneousMatrix(n): RngIntElt -> AlgMatElt
ElementaryToPowerSumMatrix(n): RngIntElt -> AlgMatElt
Bibliography
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V2.28, 13 July 2023