FUNCTION: NcsfPairing - computes the scalar product of quasi-symmetric function and noncommutative symmetric function

CALLING SEQUENCE:

NcsfPairing(qsf,ncsf)

NcsfPairing(qsf,ncsf)

NCSF[NcsfPairing](qsf,ncsf)

NCSF[NcsfPairing](qsf,ncsf)

PARAMETERS:

qsf = any quasi-symmetric function

ncsf = any noncommutative symmetric function

SYNOPSIS:

• The NcsfPairing function computes the scalar product of a quasi-symmetric function and a noncommutative symmetric function. The scalar product is defined by: <F[I],R[J]>=1 if I=J and 0 otherwise where I and J are two compositions, F is the dual basis of Ribbon Schur basis.

• qsf is any expression in terms of the basic quasi-symmetric functions.

• ncsf is any expression in terms of the basic noncommutative symmetric functions.

• The qsf and the ncsf functions are expanded. qsf (resp. ncsf) is expressed on F basis (resp. R basis). The result is not collected.

• In first argument (resp. second argument), we consider as coefficient noncommutative symmetric functions (resp. quasi-symmetric functions).

• One may add 'noexpand' just after the argument ncsf to choose not to expand qsf and ncsf before treating it.

• One may collect the result by adding an argument: this is done by NcsfPairing(qsf,ncsf,'collect').

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

EXAMPLES:

> with(NCSF):
> NcsfPairing((1+q)^5*F[3,2],R[3,2]):            # expands the input
> NcsfPairing((1+q)^5*F[3,2],R[3,2],noexpand):   # does not expand (1+q)^5
> NcsfPairing((1+q)^5*F[3,2],R[3,2],collect):    # collects the result
> NcsfPairing(F[3]*R[3],F[3]*R[3]);
 
                                   F[3] R[3]
 
> NcsfPairing((1-q)^5*F[3,2],R[3,2],noexpand,collect);
 
                                          5
                                   (1 - q)