FUNCTION: SfPlethysm - plethysm of symmetric functions

CALLING SEQUENCE:

SfPlethysm(sf_1, sf_2)

SYMF[SfPlethysm](sf_1, sf_2)

PARAMETERS:

sf_1, sf_2 = any symmetric functions

SYNOPSIS:

• The SfPlethysm function computes the plethysm sf_1[sf_2].

• One may specify that sf_1 and sf_2 are expressed on the bases b_1 and b_2 and that one wants the result to be in the b_3 basis by using either SfPlethysm(sf_1, sf_2, b_1), SfPlethysm(sf_1, sf_2, b_1, b_2) or SfPlethysm(sf_1, sf_2, b_1, b_2, b_3). The names b_1, b_2 and b_3 must be known bases.

• The plethysm operation is defined as follows: let p.i be the i-th power- sum symmetric function and sf_1, sf_2, sf_3 be any symmetric functions, then, 1. (sf_1 + sf_2)[sf_3] = sf_1[sf_3] + sf_2[sf_3] 2. (sf_1 * sf_2)[sf_3] = sf_1[sf_3] * sf_2[sf_3] 3. sf_1[p.i] = p.i[sf_1] 4. p.i[p.j] = p.(i*j)

• The default is to compute plethysms on the p-basis and return the result expressed on the p-basis.

• Special algorithms have been included to compute plethysms between elementary and complete symmetric functions in the basis of Schur functions. For that purpose, use all five arguments that is SfPlethysm(sf_1, sf_2, b_1, b_2, 's') where b_1 and b_2 are either 'e' or 'h'. These special plethysms use Newton's formula and Muir's formula. Note that these special algorithms are available when sf_2 is a single elementary, complete or Schur symmetric function with some coefficient. Otherwise, the system uses the p-basis.

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

EXAMPLES:

> with(SYMF):
> SfPlethysm(p1*s[2], h1);
 
                                               3
                             1/2 p2 p1 + 1/2 p1
 
> SfPlethysm(e2*e1, h3, 'e', 'h', 's');
 
     s[8, 1] + s[7, 2] + s[7, 1, 1] + 2 s[6, 3] + s[6, 2, 1] + s[5, 4]
 
             + 2 s[5, 3, 1] + s[4, 3, 2] + s[3, 3, 3]