FUNCTION: ToL - express any noncommutative symmetric function in the L-basis (noncommutative elementary symmetric function)

CALLING SEQUENCE:

ToL(ncsf)

ToL(ncsf, b)

NCSF[ToL](ncsf)

NCSF[ToL](ncsf, b)

PARAMETERS:

ncsf = any noncommutative symmetric function

b = any name of a known basis

SYNOPSIS:

• The ToL function computes the expansion of ncsf in the L-basis.

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

• The noncommutative symmetric function ncsf is expanded and the result is not collected.

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

• The call ToL(ncsf, 'L') does not affect the argument ncsf.

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

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

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

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

EXAMPLES:

> with(NCSF):
> ToL((1+q)^5*R[2,3]):                        # expands the input
> ToL((1+q)^5*R[2,3],noexpand):               # does not expand (1+q)^5
> ToL((1+q)^5*R[2,3],collect):                # collects the result
> ToL((1+q)^5*R[2,3],'R'):                    # specifies a basis
> ToL((1+q)^5*R[2,3],noexpand,'R',collect):   # the most efficient
> ToL((1-q)^3*S[3]+S[1,2],noexpand,'S',collect);
 
                3               3                     3
         (1 - q)  L[3] - (1 - q)  L[2, 1] + (- (1 - q)  - 1) L[1, 2]
 
                        3
              + ((1 - q)  + 1) L[1, 1, 1]