FUNCTION: TableYY - table of all double Schubert polynomials

CALLING SEQUENCE:

TableYY(n)

TableYY(n, perm)

SP[TableYY](n)

SP[TableYY](n, perm)

PARAMETERS:

n = any positive integer denoting the degree of a symmetric group

perm = any list denoting a permutation

SYNOPSIS:

• The TableYY function returns the table of all double Schubert polynomials indexed by codes of permutations in Sn. The polynomials are expressed in the basis of monomials.

• When called with the second argument perm, the second alphabet (the yi's) is specialized as the permutation perm of the first one (the xi's) : in other words, y.i is sent to x.perm[i].

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

EXAMPLES:

> with(SP):
> t:=TableYY(3);
 
              t := table([
                       [0, 0, 0] = 1
                       [1, 0, 0] = x1 - y1
                       [2, 0, 0] = (x1 - y2) (x1 - y1)
                       [0, 1, 0] = x2 - y2 + x1 - y1
                       [2, 1, 0] = (x2 - y1) (x1 - y2) (x1 - y1)
                       [1, 1, 0] = (x2 - y1) (x1 - y1)
                   ])
 
> t[ [0,1,0] ];
 
                             x2 - y2 + x1 - y1
 
> TableYY(3, [3,2,1]);
 
                table([
                    [0, 0, 0] = 1
                    [1, 0, 0] = x1 - x3
                    [2, 0, 0] = (x1 - x2) (x1 - x3)
                    [0, 1, 0] = x1 - x3
                    [2, 1, 0] = (x2 - x3) (x1 - x2) (x1 - x3)
                    [1, 1, 0] = (x2 - x3) (x1 - x3)
                ])