FUNCTION: SfTheta - apply the theta-automorphism

CALLING SEQUENCE:

SfTheta(sf, a)

SfTheta(sf, q, t)

SYMF[SfTheta](sf, a)

SYMF[SfTheta](sf, q, t)

PARAMETERS:

sf = any symmetric function

a, q, t = any names or expressions

SYNOPSIS:

• The SfTheta function realizes a certain multiplicative automorphism of the ring of symmetric functions. It is defined on power-sum functions as follows: SfTheta(sf, a) gives the image of sf under the transformation p.i --> a*p.i.

• SfTheta(sf, q, t) is the image of sf under the transformation: p.i --> (1-q^i)/(1-t^i) * p.i.

• The result is given in the p-basis.

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

EXAMPLES:

> with(SYMF):
> SfTheta(s[4,1], a);
 
        3      2        4      3         5   5                   2
   1/6 a  p3 p1  + 1/6 a  p2 p1  + 1/30 a  p1  - 1/5 a p5 - 1/6 a  p3 p2
 
> SfTheta(p3, q, t);
 
                                         3
                                   (1 - q ) p3
                                 - -----------
                                            3
                                     - 1 + t
 
> SfTheta(s[2,1], q, t);
 
                                  3   3               3
                         (- 1 + q)  p1        (- 1 + q ) p3
                     1/3 -------------- - 1/3 -------------
                                    3                   3
                           (- 1 + t)             - 1 + t