FUNCTION: SfInternal - compute the inner tensor product of symmetric functions

CALLING SEQUENCE:

SfInternal(sf_1, sf_2, b_1, b_2)

SYMF[SfInternal](sf_1, sf_2, b_1, b_2)

PARAMETERS:

sf_1, sf_2 = any symmetric functions

b_1, b_2 = any strings denoting names of known bases

SYNOPSIS:

• The SfInternal function computes the inner tensor product of symmetric functions sf_1 and sf_2. This operation is also called the internal product. Since sf_1 and sf_2 may be seen as virtual characters, SfInternal computes the symmetric function associated with the product (Kronecker product) of these two virtual characters.

• For the basis of products of power-sums, SfInternal(...p2^j p1^i, ...p2^j' p1^i') = 0 or = SfZee([...,2^j,1^i]) ...p2^j p1^i whether (...,j,i) <> (...,j',i') or not.

• The result is expressed in the p-basis.

• In the case that sf_1 and sf_2 are solely expressed in the basis b_1 and b_2, one may specify these bases.

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

EXAMPLES:

> with(SYMF):
> SfInternal(p3*p3*p2*p1*p1, p3*p3*p2*p1*p1);
 
                                     2      2
                                72 p3  p2 p1
 
> SfZee([3,3,2,1,1]);
 
                                      72
 
> SfInternal(s[3,2,1], p3*h3, 's');
 
                                    2            3
                            - 2/3 p3  - 1/3 p3 p1