FUNCTION: SfAExpand - expand a SFA expression

CALLING SEQUENCE:

SfAExpand(sfa)

SFA[SfAExpand](sfa)

PARAMETERS:

sfa = any valid expression in SFA

SYNOPSIS:

• The SfAExpand function expands, whenever possible, any expression in SFA. It is a direct implementation of the lambda-ring structure, i.e. symmetric functions are considered as operators over the ring of polynomials.

• Expansion is made considering the shape of the alphabet of each symmetric function appearing in sfa, and also products of symmetric functions. Expansion is inductively propagated, thus enabling chains of plethysms.

• Alphabets are either formal alphabets (A1, A2, ...), or real constants (any float object or any formal variable <> Ai), or variables (when explicitely so declared), or symmetric functions of SFA, or any linear combination of alphabets. For instance, 3/2*A1 + k*(A2 - s[3](-A1)) is a valid alphabet.

• One may declare variables by setting e.g. SfAVars({ {x}, {y}, z1, z2 }). Here, z1, z2, all xi's, and all yi's will be held as variables in alphabets.

• Rules used for e.g. power sums symmetric functions are:
  p[i](-A) = -p[i](A)
  p[i](A B) = p[i](A) p[i](B)
  p[i](A+B) = p[i](A)+p[i](B)
  p[i](k) = k, k real constant (<> Ai)
  p[i](x) = x^i, x explicitely declared variable
  p[i](Ai) = p[i](Ai), Ai formal alphabet
  p[i](p[j](A)) = p[ij](A) (plethysm, from previous equations) where i and j are positive integers. Equivalent rules for the other bases are also implemented.

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

EXAMPLES:

> with(SFA):
> SfAExpand( p[4,2](7+A1) );
 
                   49 + 7 p[2](A1) + 7 p[4](A1) + p[4, 2](A1)
 
> SfAExpand( s[2](A1 - 3/2*A2 + k) );
 
            2
       1/2 k  + 1/2 k - 3/2 k s[1](A2) + k s[1](A1) + 3/8 s[2](A2)
 
        + 15/8 s[1, 1](A2) - 3/2 s[1](A1) s[1](A2) + s[2](A1)
 
> SfAVars({ {x}, {y}, z1 }):
> SfAExpand( p[3]( q*m[2,1]( x1-k+A2 ) - z1^2+C ) );
 
    3                3         6         6                              2
q x1  p[6](A2) - q x1  k - q x1  k + q x1  p[3](A2) - q k p[6](A2) + q k
 
                                                        6
  + q k - q k p[3](A2) + q p[6, 3](A2) - q p[9](A2) - z1  + C