FUNCTION: SfScalar - scalar product of symmetric functions

CALLING SEQUENCE:

SfScalar(sf_1, sf_2)

SfScalar(sf_1, sf_2, sfp)

SYMF[SfScalar](sf_1, sf_2)

SYMF[SfScalar](sf_1, sf_2, sfp)

PARAMETERS:

sf_1, sf_2 = any symmetric functions

sfp = any procedure accepting a partition as input

SYNOPSIS:

• The SfScalar function computes the scalar product between the symmetric functions sf_1 and sf_2.

• This scalar product is such that Schur functions constitute an orthonormal basis.

• The p-basis is an orthogonal basis. Let sf=...p3^i p2^j p1^k, then SfScalar(sf, sf) = SfZee([...3^i, 2^j, 1^k]).

• One can choose another scalar product, for which the p-basis will still remain orthogonal by replacing SfZee with any procedure sfp accepting partitions.

• One may specify that sf_1 is expressed on basis b_1 and sf_2 is expressed on basis b_2 by adding two arguments as follows: 1. SfScalar(sf_1, sf_2, b_1, b_2) 2. SfScalar(sf_1, sf_2, b_1, b_2, sfp)

• When specifying the bases, it uses the property that the Schur basis is orthonormal, and the monomial basis is the dual basis of the basis of products of complete symmetric functions, for the usual scalar product described previously.

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

EXAMPLES:

> with(SYMF):
> SfScalar(p3^2*p1, p3^2*p1, 'p', 'p');
 
                                       18
 
> SfZee([3,3,1]);
 
                                       18
 
> SfScalar(s[6,3,3,3] + q*s[14,3,3], s[6,3,3,3] - q*s[14,3,1], 's', 's');
 
                                        1