FUNCTION: TableX - table of all Schubert polynomials

CALLING SEQUENCE:

TableX(n)

SP[TableX](n)

PARAMETERS:

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

SYNOPSIS:

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

• When called with the second argument 'Pe', it returns the table of all Schubert polynomials in Sn, expressed in the dual basis of the monomial basis, i.e. the basis of products of elementary symmetric functions on an alphabet flag. For instance, Pe[1, 2, 0, 1, 0] stands for the product e1(x1,x2,x3,x4)*e2(x1,x2,x3)*e1(x1).

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

EXAMPLES:

> with(SP):
> t:=TableX(3);
 
                      t := table([
                               [1, 2, 3] = 1
                                             2
                               [3, 1, 2] = x1
                               [1, 3, 2] = x2 + x1
                               [2, 1, 3] = x1
                               [2, 3, 1] = x2 x1
                                                2
                               [3, 2, 1] = x2 x1
                           ])
 
> t[ [2,3,1] ];
 
                                   x2 x1
 
> TableX(3, 'Pe');
 
                   table([
                     [3, 1, 2] = Pe[1, 1, 0] - Pe[2, 0, 0]
                     [3, 2, 1] = Pe[2, 1, 0]
                     [1, 2, 3] = Pe[0]
                     [2, 3, 1] = Pe[2, 0, 0]
                     [2, 1, 3] = Pe[1, 0]
                     [1, 3, 2] = Pe[1, 0, 0]
                   ])