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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
#############################################################################
##
#F MolienSeries( [<tbl>, ]<psi>[, <chi>] )
##
## <#GAPDoc Label="MolienSeries">
## <ManSection>
## <Func Name="MolienSeries" Arg='[tbl, ]psi[, chi]'/>
##
## <Description>
## The <E>Molien series</E> of the character <M>\psi</M>,
## relative to the character <M>\chi</M>, is the rational function given by
## the series
## <M>M_{{\psi,\chi}}(z) = \sum_{{d = 0}}^{\infty} [\chi,\psi^{[d]}] z^d</M>,
## where <M>\psi^{[d]}</M> denotes the symmetrization of <M>\psi</M>
## with the trivial character of the symmetric group <M>S_d</M>
## (see <Ref Func="SymmetricParts"/>).
## <P/>
## <Ref Func="MolienSeries"/> returns the Molien series of <A>psi</A>,
## relative to <A>chi</A>, where <A>psi</A> and <A>chi</A> must be
## characters of the same character table;
## this table must be entered as <A>tbl</A> if <A>chi</A> and <A>psi</A>
## are only lists of character values.
## The default for <A>chi</A> is the trivial character of <A>tbl</A>.
## <P/>
## The return value of <Ref Func="MolienSeries"/> stores a value for the
## attribute <Ref Attr="MolienSeriesInfo"/>.
## This admits the computation of coefficients of the series with
## <Ref Func="ValueMolienSeries"/>.
## Furthermore, this attribute gives access to numerator and denominator
## of the Molien series viewed as rational function,
## where the denominator is a product of polynomials of the form
## <M>(1-z^r)^k</M>; the Molien series is also displayed in this form.
## Note that such a representation is not unique, one can use
## <Ref Func="MolienSeriesWithGivenDenominator"/>
## to obtain the series with a prescribed denominator.
## <P/>
## For more information about Molien series, see <Cite Key="NPP84"/>.
## <P/>
## <Example><![CDATA[
## gap> t:= CharacterTable( AlternatingGroup( 5 ) );;
## gap> psi:= First( Irr( t ), x -> Degree( x ) = 3 );;
## gap> mol:= MolienSeries( psi );
## ( 1-z^2-z^3+z^6+z^7-z^9 ) / ( (1-z^5)*(1-z^3)*(1-z^2)^2 )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MolienSeries" );
#############################################################################
##
#F MolienSeriesWithGivenDenominator( <molser>, <list> )
##
## <#GAPDoc Label="MolienSeriesWithGivenDenominator">
## <ManSection>
## <Func Name="MolienSeriesWithGivenDenominator" Arg='molser, list'/>
##
## <Description>
## is a Molien series equal to <A>molser</A> as rational function,
## but viewed as quotient with denominator
## <M>\prod_{{i = 1}}^n (1-z^{{r_i}})</M>,
## where <M><A>list</A> = [ r_1, r_2, \ldots, r_n ]</M>.
## If <A>molser</A> cannot be represented this way,
## <K>fail</K> is returned.
## <P/>
## <Example><![CDATA[
## gap> MolienSeriesWithGivenDenominator( mol, [ 2, 6, 10 ] );
## ( 1+z^15 ) / ( (1-z^10)*(1-z^6)*(1-z^2) )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MolienSeriesWithGivenDenominator" );
##############################################################################
##
#A MolienSeriesInfo( <ratfun> )
##
## <#GAPDoc Label="MolienSeriesInfo">
## <ManSection>
## <Attr Name="MolienSeriesInfo" Arg='ratfun'/>
##
## <Description>
## If the rational function <A>ratfun</A> was constructed by
## <Ref Func="MolienSeries"/>,
## a representation as quotient of polynomials is known such that the
## denominator is a product of terms of the form <M>(1-z^r)^k</M>.
## This information is encoded as value of <Ref Attr="MolienSeriesInfo"/>.
## Additionally, there is a special <Ref Oper="PrintObj"/> method
## for Molien series based on this.
## <P/>
## <Ref Attr="MolienSeriesInfo"/> returns a record that describes the
## rational function <A>ratfun</A> as a Molien series.
## The components of this record are
##
## <List>
## <Mark><C>numer</C></Mark>
## <Item>
## numerator of <A>ratfun</A> (in general a multiple of the numerator
## one gets by <Ref Attr="NumeratorOfRationalFunction"/>),
## </Item>
## <Mark><C>denom</C></Mark>
## <Item>
## denominator of <A>ratfun</A> (in general a multiple of the
## denominator one gets by <Ref Attr="NumeratorOfRationalFunction"/>),
## </Item>
## <Mark><C>ratfun</C></Mark>
## <Item>
## the rational function <A>ratfun</A> itself,
## </Item>
## <Mark><C>numerstring</C></Mark>
## <Item>
## string corresponding to the polynomial <C>numer</C>,
## expressed in terms of <C>z</C>,
## </Item>
## <Mark><C>denomstring</C></Mark>
## <Item>
## string corresponding to the polynomial <C>denom</C>,
## expressed in terms of <C>z</C>,
## </Item>
## <Mark><C>denominfo</C></Mark>
## <Item>
## a list of the form <M>[ [ r_1, k_1 ], \ldots, [ r_n, k_n ] ]</M>
## such that <C>denom</C> is
## <M>\prod_{{i = 1}}^n (1-z^{{r_i}})^{{k_i}}</M>.
## </Item>
## <Mark><C>summands</C></Mark>
## <Item>
## a list of records, each with the components <C>numer</C>, <C>r</C>,
## and <C>k</C>,
## describing the summand <C>numer</C><M> / (1-z^r)^k</M>,
## </Item>
## <Mark><C>pol</C></Mark>
## <Item>
## a list of coefficients, describing a final polynomial which is added
## to those described by <C>summands</C>,
## </Item>
## <Mark><C>size</C></Mark>
## <Item>
## the order of the underlying matrix group,
## </Item>
## <Mark><C>degree</C></Mark>
## <Item>
## the degree of the underlying matrix representation.
## </Item>
## </List>
## <P/>
## <Example><![CDATA[
## gap> HasMolienSeriesInfo( mol );
## true
## gap> MolienSeriesInfo( mol );
## rec( degree := 3,
## denom := x_1^12-2*x_1^10-x_1^9+x_1^8+x_1^7+x_1^5+x_1^4-x_1^3-2*x_1^2\
## +1, denominfo := [ 5, 1, 3, 1, 2, 2 ],
## denomstring := "(1-z^5)*(1-z^3)*(1-z^2)^2",
## numer := -x_1^9+x_1^7+x_1^6-x_1^3-x_1^2+1,
## numerstring := "1-z^2-z^3+z^6+z^7-z^9", pol := [ ],
## ratfun := ( 1-z^2-z^3+z^6+z^7-z^9 ) / ( (1-z^5)*(1-z^3)*(1-z^2)^2 ),
## size := 60,
## summands := [ rec( k := 1, numer := [ -24, -12, -24 ], r := 5 ),
## rec( k := 1, numer := [ -20 ], r := 3 ),
## rec( k := 2, numer := [ -45/4, 75/4, -15/4, -15/4 ], r := 2 ),
## rec( k := 3, numer := [ -1 ], r := 1 ),
## rec( k := 1, numer := [ -15/4 ], r := 1 ) ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "MolienSeriesInfo", IsRationalFunction );
#############################################################################
##
#F CoefficientTaylorSeries( <numer>, <r>, <k>, <i> )
##
## <ManSection>
## <Func Name="CoefficientTaylorSeries" Arg='numer, r, k, i'/>
##
## <Description>
## is the coefficient of <M>z^<A>i</A></M> in the Taylor series expansion of
## the quotient of polynomials
## <M>p(z) / ( 1 - z^{<A>r</A>} )^{<A>k</A>}</M>,
## where <A>numer</A> is the coefficients list of the numerator polynomial
## <M>p(z)</M>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "CoefficientTaylorSeries" );
#############################################################################
##
#F SummandMolienSeries( <tbl>, <psi>, <chi>, <i> )
##
## <ManSection>
## <Func Name="SummandMolienSeries" Arg='tbl, psi, chi, i'/>
##
## <Description>
## is the summand of the Molien series of the character table <A>tbl</A>,
## for the characters <A>psi</A> and <A>chi</A>, that corresponds to class
## <A>i</A>.
## That is, the returned value is the quotient
## <Display Mode="M">
## \chi(g) \cdot \det(D(g)) / \det(z I - D(g))
## </Display>
## where <M>g</M> is in class <A>i</A>, <M>D</M> is a representation with
## character <A>psi</A>, and <M>z</M> is the indeterminate.
## <P/>
## The result is a record with components <C>numer</C> and <C>a</C>,
## with the following meaning.
## <P/>
## Write the denominator as a product of cyclotomic polynomials,
## encode this as a list <C>a</C> where at position <M>r</M> the
## multiplicity of the <M>r</M>-th cyclotomic polynomial <M>\Phi_r</M>
## is stored.
## (For that, we possibly must change the numerator.)
## We get
## <Display Mode="M">
## 1 / \det(z I - D(g))
## = P(z) / \left( \prod_{{d \mid n}} \Phi_d^{a_d}(z) \right) .
## </Display>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "SummandMolienSeries" );
#############################################################################
##
#F ValueMolienSeries( <molser>, <i> )
##
## <#GAPDoc Label="ValueMolienSeries">
## <ManSection>
## <Func Name="ValueMolienSeries" Arg='molser, i'/>
##
## <Description>
## is the <A>i</A>-th coefficient of the Molien series <A>series</A>
## computed by <Ref Func="MolienSeries"/>.
## <P/>
## <Example><![CDATA[
## gap> List( [ 0 .. 20 ], i -> ValueMolienSeries( mol, i ) );
## [ 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 4, 1, 5, 1, 6, 1, 7 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ValueMolienSeries" );
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