EtaQuotient(C, L)¶
EtaQuotient implements the (multiplicative) group of eta-functions in their expansion at infinity. Elements can be represented as Laurent series in q
with a prefactor of q^
(n/24
), (n=0
,…,23). Note that this domain keeps the “fractional part” always separate from the “series part” even when the fractional part is an integer.
- 1: %
- from MagmaWithUnit
- *: (%, %) -> %
- from Magma
- /: (%, %) -> %
- from Group
- =: (%, %) -> Boolean
- from BasicType
- ^: (%, Integer ) -> %
- from Group
- ^: (%, NonNegativeInteger ) -> %
- from MagmaWithUnit
- ^: (%, PositiveInteger ) -> %
- from Magma
- ~=: (%, %) -> Boolean
- from BasicType
- coerce: % -> OutputForm
- from CoercibleTo OutputForm
- coerce: C -> %
coerce(c)
returns an element of this domain corresponding toc
.- commutator: (%, %) -> %
- from Group
- conjugate: (%, %) -> %
- from Group
- eta: PositiveInteger -> %
eta(n)
returnsq^
(n/24
)*prod_{k=1
}^infty (1-q^
{kn
}).
- etaPower: (PositiveInteger , Integer ) -> %
etaPower(d, e)
returns eta(d
)^e.
- etaQuotient: (List Integer , List Integer ) -> %
etaQuotient(divs, r)
returns the $eta$-quotient product(eta(divs
.i
)^(r
.i
),i=1
..#divs). It is assumed that the lengths of the input lists are equal.- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState , %) -> HashState
- from SetCategory
- inv: % -> %
- from Group
- latex: % -> String
- from SetCategory
- leftPower: (%, NonNegativeInteger ) -> %
- from MagmaWithUnit
- leftPower: (%, PositiveInteger ) -> %
- from Magma
- leftRecip: % -> Union(%, failed)
- from MagmaWithUnit
- one?: % -> Boolean
- from MagmaWithUnit
- prefactor: % -> NonNegativeInteger
prefactor(x)
returns the exponent times 24 of the fractionalq
power ofx
whereq
is the variable ofL
. The returned value is in the range 0..23, because integer powers ofq
are moved to the series part.
- q24: Integer -> %
q24(n)
returnsq^
(n/24
).- recip: % -> Union(%, failed)
- from MagmaWithUnit
- retract: % -> L
retract(x)
aborts with error if prefactor is not divisible by 24. Otherwise it returns q^r*series(x
) where r=prefactor(x
)/24
andq
is the variable ofL
.- rightPower: (%, NonNegativeInteger ) -> %
- from MagmaWithUnit
- rightPower: (%, PositiveInteger ) -> %
- from Magma
- rightRecip: % -> Union(%, failed)
- from MagmaWithUnit
- sample: %
- from MagmaWithUnit
- series: % -> L
series(x)
returns the Laurent series part ofx
.x
=q24
(prefactorx
)*series(x
)
- toEta: Polynomial C -> %
toEta(p)
assumes that all variables are of the formEi
with the letterE
and a positive numberi
. In the polynomialp
the powers Ei^ni will be replaced by eta(i
)^ni. Ifl
is the leading monomialp
, then it is assumed that prefactor(toEta(l
))=prefactor(toEta(m
)) for every monomialm
ofp
. If