QEtaSpecificationRing C¶
qetaspecring.spad line 91 [edit on github]
QEtaSpecificationRing is the domain of integer linear combinations of eta-quotients.
- 0: %
- from AbelianMonoid 
- 1: %
- from MagmaWithUnit 
- *: (%, %) -> %
- from Magma 
- *: (%, C) -> %
- from RightModule C 
- *: (C, %) -> %
- from LeftModule C 
- *: (Integer, %) -> %
- from AbelianGroup 
- *: (NonNegativeInteger, %) -> %
- from AbelianMonoid 
- *: (PositiveInteger, %) -> %
- from AbelianSemiGroup 
- +: (%, %) -> %
- from AbelianSemiGroup 
- -: % -> %
- from AbelianGroup 
- -: (%, %) -> %
- from AbelianGroup 
- ^: (%, NonNegativeInteger) -> %
- from MagmaWithUnit 
- ^: (%, PositiveInteger) -> %
- from Magma 
- annihilate?: (%, %) -> Boolean
- from Rng 
- antiCommutator: (%, %) -> %
- associator: (%, %, %) -> %
- from NonAssociativeRng 
- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing 
- charthRoot: % -> Union(%, failed) if C has CharacteristicNonZero
- coefficient: (%, QEtaSpecification) -> C
- from FreeModuleCategory(C, QEtaSpecification) 
- coefficients: % -> List C
- from FreeModuleCategory(C, QEtaSpecification) 
- coerce: % -> %
- from Algebra % 
- coerce: % -> OutputForm
- from CoercibleTo OutputForm 
- coerce: C -> %
- from Algebra C 
- coerce: Fraction Polynomial C -> % if C has IntegralDomain
- coerce(p)returns specificationPolynomial(- p,- e) where- eis the first letter of the first variable of- p. This, of course, assumes that all variables of- pare of the form ed or ed_g where- dis a positive integer and 0<g<=d/2
- coerce: Integer -> %
- from NonAssociativeRing 
- coerce: List Record(k: QEtaSpecification, c: C) -> %
- from MonoidRingCategory(C, QEtaSpecification) 
- coerce: Polynomial C -> %
- coerce(p)returns specificationPolynomial(- p,- e) where- eis the first letter of the first variable of- p. This, of course, assumes that all variables of- pare of the form ed or ed_g where- dis a positive integer and 0<g<=d/2
- coerce: QEtaSpecification -> %
- commutator: (%, %) -> %
- from NonAssociativeRng 
- construct: List Record(k: QEtaSpecification, c: C) -> %
- from IndexedProductCategory(C, QEtaSpecification) 
- constructOrdered: List Record(k: QEtaSpecification, c: C) -> %
- from IndexedProductCategory(C, QEtaSpecification) 
- convert: % -> InputForm if C has Finite and QEtaSpecification has Finite
- from ConvertibleTo InputForm 
- etaPolynomial: % -> Polynomial C
- etaPolynomial(x)returns etaPolynomial(- x,”E”,- "Y").
- etaPolynomial: (%, String, String) -> Polynomial C
- etaPolynomial(x,e,y)replaces in each term of- xthe specification spec by monomial(spec,- e,- y).
- etaRationalFunction: % -> Fraction Polynomial C if C has IntegralDomain
- etaRationalFunction(x)returns etaRationalFunction(- x,”E”).
- etaRationalFunction: (%, String) -> Fraction Polynomial C if C has IntegralDomain
- etaRationalFunction(x,e)replaces in each term of- xthe specification spec by quotient(spec,- e).
- hash: % -> SingleInteger if C has Finite and QEtaSpecification has Finite
- from Hashable 
- hashUpdate!: (HashState, %) -> HashState if C has Finite and QEtaSpecification has Finite
- from Hashable 
- index: PositiveInteger -> % if C has Finite and QEtaSpecification has Finite
- from Finite 
- latex: % -> String
- from SetCategory 
- leadingCoefficient: % -> C
- from IndexedProductCategory(C, QEtaSpecification) 
- leadingMonomial: % -> %
- from IndexedProductCategory(C, QEtaSpecification) 
- leadingTerm: % -> Record(k: QEtaSpecification, c: C)
- from IndexedProductCategory(C, QEtaSpecification) 
- leftPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit 
- leftPower: (%, PositiveInteger) -> %
- from Magma 
- leftRecip: % -> Union(%, failed)
- from MagmaWithUnit 
- level: % -> PositiveInteger
- level(x)returns the least common multiple of the levels of all terms.
- linearExtend: (QEtaSpecification -> C, %) -> C
- from FreeModuleCategory(C, QEtaSpecification) 
- listOfTerms: % -> List Record(k: QEtaSpecification, c: C)
- lookup: % -> PositiveInteger if C has Finite and QEtaSpecification has Finite
- from Finite 
- map: (C -> C, %) -> %
- from IndexedProductCategory(C, QEtaSpecification) 
- modularGamma0?: % -> Boolean
- modularGamma0?(x)returns- trueif the specification in each term of- xis modular with respect to $Gamma_0(- N)$ where $- N$ is the- lcmof all levels of the specifications of the terms of- x.
- modularGamma1?: % -> Boolean
- modularGamma1?(x)returns- trueif the specification in each term of- xis modular with respect to $Gamma_0(- N)$ where $- N$ is the- lcmof all levels of the specifications of the terms of- x.
- monomial?: % -> Boolean
- from IndexedProductCategory(C, QEtaSpecification) 
- monomial: (C, QEtaSpecification) -> %
- from IndexedProductCategory(C, QEtaSpecification) 
- monomials: % -> List %
- from FreeModuleCategory(C, QEtaSpecification) 
- one?: % -> Boolean
- from MagmaWithUnit 
- opposite?: (%, %) -> Boolean
- from AbelianMonoid 
- plenaryPower: (%, PositiveInteger) -> %
- from NonAssociativeAlgebra C 
- pretty: (%, Integer) -> OutputForm
- pretty(x,fmt)returns pretty(- x,- fmt,0).
- pretty: (%, Integer, (Integer, Integer) -> OutputForm, Fraction Integer) -> OutputForm
- pretty(x,fmt,v,e)multiplies- xby q^e and transforms the relation into a nicely looking output form using the variable function- vdeciding via format whether it must print a factor of- q.
- pretty: (%, Integer, Fraction Integer) -> OutputForm
- pretty(x,fmt,e)chooses an appropriate variable function- vaccording to- fmtand returns pretty(- x,- fmt,- v,- e).
- purify: % -> %
- purify(x)applies purify(spec) to each term c*spec of- x.
- random: () -> % if C has Finite and QEtaSpecification has Finite
- from Finite 
- recip: % -> Union(%, failed)
- from MagmaWithUnit 
- reductum: % -> %
- from IndexedProductCategory(C, QEtaSpecification) 
- retract: % -> C
- from RetractableTo C 
- retract: % -> QEtaSpecification
- retractIfCan: % -> Union(C, failed)
- from RetractableTo C 
- retractIfCan: % -> Union(QEtaSpecification, failed)
- rightPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit 
- rightPower: (%, PositiveInteger) -> %
- from Magma 
- rightRecip: % -> Union(%, failed)
- from MagmaWithUnit 
- sample: %
- from AbelianMonoid 
- size: () -> NonNegativeInteger if C has Finite and QEtaSpecification has Finite
- from Finite 
- smaller?: (%, %) -> Boolean if C has Comparable
- from Comparable 
- specificationPolynomial: (Fraction Polynomial C, String) -> % if C has IntegralDomain
- specificationPolynomial(p,e)replaces each variable- eiby the respective eta-specification and thus turns- pinto an element of this domain. Note that the specifications of each monomial may live in different levels. This function can be seen as a coercion.
- specificationPolynomial: (Polynomial C, String) -> %
- specificationPolynomial(p,e)replaces each variable- eiby the respective eta-specification and thus turns- pinto an element of this domain. Note that the specifications of each monomial may live in different levels. This function can be seen as a coercion.
- specificationPolynomial: (Polynomial C, String, List QEtaSpecification) -> %
- specificationPolynomial(p,m,specs)replaces each of the variables- miin- pby the respective element- specs.- iand thus creates a formal linear combination of eta-quotients.
- specificationPolynomial: (PositiveInteger, Fraction Polynomial C, String) -> % if C has IntegralDomain
- specificationPolynomial(nn,p,e)replaces each variable- eiby the respective eta-specification (of level- nn) and thus turns- pinto an element of this domain. It can be seen as a coercion function.
- specificationPolynomial: (PositiveInteger, Polynomial C, String) -> %
- specificationPolynomial(nn,p,e)replaces each variable- eiby the respective eta-specification (of level- nn) and thus turns- pinto an element of this domain. It can be seen as a coercion function.
- subtractIfCan: (%, %) -> Union(%, failed)
- support: % -> List QEtaSpecification
- from FreeModuleCategory(C, QEtaSpecification) 
- terms: % -> List Record(k: QEtaSpecification, c: C)
- from MonoidRingCategory(C, QEtaSpecification) 
- zero?: % -> Boolean
- from AbelianMonoid 
Algebra %
Algebra C
BiModule(%, %)
BiModule(C, C)
CharacteristicNonZero if C has CharacteristicNonZero
CharacteristicZero if C has CharacteristicZero
CoercibleFrom QEtaSpecification
Comparable if C has Comparable
ConvertibleTo InputForm if C has Finite and QEtaSpecification has Finite
Finite if C has Finite and QEtaSpecification has Finite
FreeModuleCategory(C, QEtaSpecification)
Hashable if C has Finite and QEtaSpecification has Finite
IndexedDirectProductCategory(C, QEtaSpecification)
IndexedProductCategory(C, QEtaSpecification)
Module %
Module C
MonoidRingCategory(C, QEtaSpecification)