FUNCTION: SfAOmega - apply the omega-automorphism

CALLING SEQUENCE:

SfAOmega(sfa)

SfAOmega(sfa, alist)

SFA[SfAOmega](sfa)

SFA[SfAOmega](sfa, alist)

PARAMETERS:

sfa = any valid expression in SFA

alist = a list of alphabets

SYNOPSIS:

• The omega-automorphism is an involution defined on symmetric functions by:
  h[I](A) <-> e[I](A)
  s[I](A) <-> s[I~](A)
  p[I](A) <-> -1^(length(I)+|I|) p[I](A), where I~ is the conjugate partition of I, and |I| is the weight of I.

• One can apply SfAOmega solely on symmetric functions over the alphabets given in the second parameter alist.

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

EXAMPLES:

> with(SFA):
> SfAOmega( p[3,1](A1) - q*s[3](A2) );
 
                         p[3, 1](A1) - q s[1, 1, 1](A2)
 
> SfAOmega( p[3,1](A1) - q*s[3](A2), [ A1 ]);
 
                            p[3, 1](A1) - q s[3](A2)