FUNCTION: Toc - express any symmetric function in the basis of cycle indices

CALLING SEQUENCE:

Toc(sf)

Toc(sf, b)

SYMF[Toc](sf)

SYMF[Toc](sf, b)

PARAMETERS:

sf = any symmetric function

b = any name of a known basis

SYNOPSIS:

• The Toc function expresses any symmetric function in the basis of cycle indices, i.e. as a linear combination of cycle indices c[part] of conjugacy classes corresponding to the partition part=[...,3^i,2^j,1^k]: c[part] = convert(part,`+`)! / SfZee(part) * ... p3^i p2^j p1^k In other words, the c-basis is proportional to the p-basis.

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

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

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

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

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

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

EXAMPLES:

> with(SYMF):
> Toc((1+q)^3*e3*e2):                            # expands the input
> Toc((1+q)^3*e3*e2,noexpand):                   # does not expand (1+q)^3
> Toc((1+q)^3*e3*e2,collect):                    # collects the result
> Toc((1+q)^3*e3*e2,noexpand,'e'):               # the most efficient
> Toc((1+q)^3*e3*e2,'e',collect):                # specifies a basis
> Toc((1+q)^3*e3*e2,noexpand,'e',collect):
> Toc(c[2,1]*h4 - 2*q*p3);
 
     1/840 c[4, 2, 1] + 1/420 c[3, 2, 1, 1] + 1/280 c[2, 2, 2, 1]
 
     + 1/140 c[2, 2, 1, 1, 1] + 1/168 c[2, 1, 1, 1, 1, 1] - c[3] q