68.12 Molien Series

The Molien series of the character y, relative to the character c, is the rational function given by the series

MolienSeries returs the Molien series of psi, relative to chi, where psi and chi must be characters of the same character table, which must be entered as tbl if chi and psi are only lists of character values. The default for chi is the trivial character of tbl.

The return value of MolienSeries stores a value for the attribute MolienSeriesInfo (see MolienSeriesInfo). This admits the computation of coefficients of the series with ValueMolienSeries (see 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 (1-z^r)^k; the Molien series is also displayed in this form. Note that such a representation is not unique, one can use MolienSeriesWithGivenDenominator (see MolienSeriesWithGivenDenominator) to obtain the series with a prescribed denominator.

For more information about Molien series, see NPP84.

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 )                       

If the rational function ratfun was constructed by MolienSeries (see MolienSeries), a representation as quotient of polynomials is known such that the denominator is a product of terms of the form (1-z^r)^k. This information is encoded as value of MolienSeriesInfo. Additionally, there is a special PrintObj method for Molien series based on this.

MolienSeriesInfo returns a record that describes the rational function ratfun as a Molien series. The components of this record are

numer

numerator of ratfun (in general a multiple of the numerator one gets by NumeratorOfRationalFunction),

denom

denominator of ratfun (in general a multiple of the denominator one gets by NumeratorOfRationalFunction),

ratfun

the rational function ratfun itself,

numerstring

string corresponding to the polynomial numer, expressed in terms of z,

denomstring

string corresponding to the polynomial denom, expressed in terms of z,

denominfo

a list of the form [ [ r₁, k₁ ], ¼, [ r_n, k_n ] ] such that denom is Õ_{i = 1}ⁿ (1-z^r_i)^k_i.

summands

a list of records, each with the components numer, r, and k, describing the summand numer / (1-z^r)^k,

size

the order of the underlying matrix group,

degree

the degree of the underlying matrix representation.

gap> HasMolienSeriesInfo( mol );
true
gap> MolienSeriesInfo( mol );   
rec( summands := [ rec( numer := [ -24, -12, -24 ], r := 5, k := 1 ), 
      rec( numer := [ -20 ], r := 3, k := 1 ), 
      rec( numer := [ -45/4, 75/4, -15/4, -15/4 ], r := 2, k := 2 ), 
      rec( numer := [ -1 ], r := 1, k := 3 ), 
      rec( numer := [ -15/4 ], r := 1, k := 1 ) ], size := 60, degree := 3, 
  numer := 1-x_1^2-x_1^3+x_1^6+x_1^7-x_1^9, 
  denom := 1-2*x_1^2-x_1^3+x_1^4+x_1^5+x_1^7+x_1^8-x_1^9-2*x_1^10+x_1^12, 
  denominfo := [ 5, 1, 3, 1, 2, 2 ], numerstring := "1-z^2-z^3+z^6+z^7-z^9", 
  denomstring := "(1-z^5)*(1-z^3)*(1-z^2)^2", 
  ratfun := ( 1-z^2-z^3+z^6+z^7-z^9 ) / ( (1-z^5)*(1-z^3)*(1-z^2)^2 ) )

is the i-th coefficient of the Molien series series computed by MolienSeries.

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 ]

is a Molien series equal to molser as rational function, but viewed as quotient with denominator Õ_{i = 1}ⁿ (1-z^r_i), where list = [ r₁, r₂, ¼, r_n ]. If molser cannot be represented this way, fail is returned.

gap> MolienSeriesWithGivenDenominator( mol, [ 2, 6, 10 ] );       
( 1+z^15 ) / ( (1-z^10)*(1-z^6)*(1-z^2) )

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