FUNCTION: Sf2Table - convert a symmetric function into a table

CALLING SEQUENCE:

Sf2Table(sf, b)

SYMF[Sf2Table](sf, b)

PARAMETERS:

sf = any symmetric function

b = any name of a known basis

SYNOPSIS:

• The Sf2Table function converts a symmetric function into a table. The table is indexed by partitions (possibly empty) and each entry that corresponds to a partition part is interpreted as the coefficient of b[part] in the expression of the symmetric function in the basis b.

• Note that for instance if sf=s[3,1,1] and b='s' then the corresponding partition is [3,1,1] and if sf=h3*h1^2 then the corresponding partition is [3,1,1] too.

• This function can be useful if one wants to act either on coefficients or on partitions. For instance, one can use the map function to act on coefficients.

• Whenever there is a conflict between the function name Sf2Table and another name used in the same session, use the long form SYMF['Sf2Table'].

EXAMPLES:

> with(SYMF):
> t := Sf2Table(q^2*s[3,1,1] + q*s[4,2] + q + s[4,2], 's');
 
                         t := table([
                                  [4, 2] = q + 1
                                               2
                                  [3, 1, 1] = q
                                  [] = q
                              ])
 
> t := map(coefficient -> coefficient - 1, t);
 
                       t := table([
                                [4, 2] = q
                                             2
                                [3, 1, 1] = q  - 1
                                [] = q - 1
                            ])
 
> Table2Sf(t, 's');
 
                              2
                q s[4, 2] + (q  - 1) s[3, 1, 1] + (q - 1) s[]
 
> Sf2Table(q*h1^3*h2 - q^2*h4*h3^2*h1, 'h');
 
                           table([
                                                 2
                               [4, 3, 3, 1] = - q
                               [2, 1, 1, 1] = q
                           ])