FUNCTION: SfAddBasis - addition of a new basis

CALLING SEQUENCE:

SfAddBasis(b)

SfAddBasis(b, scal)

SfAddBasis(b, scal, leadcoeff)

SYMF[SfAddBasis](b)

SYMF[SfAddBasis](b, scal)

SYMF[SfAddBasis](b, scal, leadcoeff)

PARAMETERS:

b = any name

scal, leadcoeff = any procedures accepting a partition as input

SYNOPSIS:

• The function part -> scal(part) choosen by the user defines a scalar product <,> on symmetric functions for which power sums are orthogonal, and such that <p[part], p[part]> = scal(part). The usual scalar product SfScalar corresponds to scal(part) = SfZee(part).

• There exists a unique basis b[] of the linear space of symmetric functions that is orthogonal and satisfies: b[part] = leadcoeff(part)* m[part] + combination of m[mu], mu<part (with respect to the total order on partitions listed with the ListPart function).

• By default, the leading coefficient is set equal to 1, i.e. is the function which returns 1 whatever is its argument: proc(part) 1 end.

• The new basis b is added to the list of bases and will be recognized in any expression; a new procedure Tob is available, with the same syntax as for instance Tom, Tos, and is devoted to express any symmetric function in the b-basis.

• It has also the effect to define a Isb function that tests whether an expression is an element of the b-basis.

• The result is the current set of known bases.

• An error is produced if 'b' is the name of a previously defined basis.

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

EXAMPLES:

> with(SYMF):
> NewZee:=proc(part) SfZee(part, 'q', 't') end:
> SfAddBasis(Mac, NewZee);        # Macdonald polynomials, up to constants
 
                            {p, s, e, h, m, c, Mac}
 
> map(factor, Tom(Mac[2]*s[2,1], 'collect'));
 
            (2 t q + t - q - 2) m[3, 2]     (2 t q + t - q - 2) m[3, 1, 1]
  m[4, 1] + --------------------------- + 2 ------------------------------
                      t q - 1                           t q - 1
 
             (3 t q + 2 t - 2 q - 3) m[2, 2, 1]
         + 2 ----------------------------------
                           t q - 1
 
           (9 t + 11 t q - 9 q - 11) m[2, 1, 1, 1]
         + ---------------------------------------
                           t q - 1
 
              (- 1 + t) (q + 1) m[1, 1, 1, 1, 1]
         + 20 ----------------------------------
                            t q - 1
 
> map(factor, ToMac(Mac[2]*s[1], 'collect'));
 
                                     2
                         (q + 1) (q t  - 1) (- 1 + q) Mac[2, 1]
                Mac[3] + --------------------------------------
                                    2
                                  (q  t - 1) (t q - 1)
 
> map(factor, SfMat(3, 'Mac', 'm'));
 
                  [                 2                   2 ]
                  [       (q + 1) (q  + q + 1) (- 1 + t)  ]
                  [ 1  0  ------------------------------- ]
                  [               2                       ]
                  [             (q  t - 1) (t q - 1)      ]
                  [                                       ]
                  [        (2 t q + t + q + 2) (- 1 + t)  ]
                  [ 0  1   -----------------------------  ]
                  [                      2                ]
                  [                   q t  - 1            ]
                  [                                       ]
                  [ 0  0                 1                ]