FUNCTION: ToE - express any quasi-symmetric function in the E-basis (dual basis of noncommutative elementary symmetric function)

CALLING SEQUENCE:

ToE(qsf)

ToE(qsf, b)

NCSF[ToE](qsf)

NCSF[ToE](qsf, b)

PARAMETERS:

qsf = any quasi-symmetric function

b = any name of a known basis

SYNOPSIS:

• The ToE function computes the expansion of qsf in the E-basis.

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

• The quasi-symmetric function qsf is expanded and the result is not collected.

• One may specify by a second argument, say b, that qsf is solely expressed in terms of the known basis b.

• The call ToE(qsf, 'E') does not affect the argument qsf.

• One may add 'noexpand' just after the argument qsf to choose not to expand the quasi-symmetric function qsf before treating it.

• One may collect the result by adding a third argument: this is done by ToE(qsf, b, 'collect'). For instance, ToE(qsf, 'E', 'collect') may be used to collect the argument qsf.

• The noncommutative multiplication is denoted by the &* operator.

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

EXAMPLES:

> with(NCSF):
> ToE((1+q)^2*F[2,1]*F[2]):                      # expands the input
> ToE((1+q)^2*F[2,1]*F[2],noexpand):             # does not expand (1+q)^2
> ToE((1+q)^2*F[2,1]*F[2],collect):              # collects the result
> ToE((1+q)^5*E[2,3]*E[1,2],'E'):                # specifies a basis
> ToE((1+q)^2*F[2,1]*F[2],noexpand,'F',collect); # the most efficient
> ToE((1-q)^2*F[1,2,1]+QPh[2]&*F[1,1],noexpand,collect);
 
                         2                         2
   - E[1, 2, 1] + (1 - q)  E[1, 1, 1, 1] + ((1 - q)  - 2) E[2, 2]
 
                   2                             2
   + (- 1 + (1 - q) ) E[2, 1, 1] + (- 1 + (1 - q) ) E[1, 1, 2] - E[4]
 
   - E[1, 3] - E[3, 1]