FUNCTION: Sf2Char - convert a symmetric function into a virtual character

CALLING SEQUENCE:

Sf2Char(sf)

SYMF[Sf2Char](sf)

PARAMETERS:

sf = any symmetric function

SYNOPSIS:

• The Sf2Char function converts a symmetric function into a virtual character. This function is the inverse of Char2Sf.

• The Sf2Char function realizes the isomorphism between the linear space of homogeneous symmetric functions of degree n and class functions on the symmetric group of degree n (Frobenius isomorphism).

• The result is a linear combination of cl[part] where part is a partition.

• Whenever there is a conflict between the function name Sf2Char and another name used in the same session, use the long form SYMF['Sf2Char'].

EXAMPLES:

> with(SYMF):
> Sf2Char(s[3,1,1]);
 
              - 2 cl[2, 2, 1] + 6 cl[1, 1, 1, 1, 1] + cl[5]
 
> Sf2Char(p3*s[3] - e4);
 
      3 cl[3, 2, 1] - cl[1, 1, 1, 1] + cl[4] - cl[3, 1] + cl[2, 1, 1]
 
              - cl[2, 2] + 6 cl[3, 3] + 3 cl[3, 1, 1, 1]