FUNCTION: NcaYang - compute a special element of the nilCoxeter algebra

CALLING SEQUENCE:

NcaYang(perm)

NCA[NcaYang](perm)

PARAMETERS:

perm = any list denoting a permutation

SYNOPSIS:

• The NcaYang function calculates a special element of the nilCoxeter algebra.

• { NcaYang(perm), perm in ListPerm(n) } is a linear basis of the nilCoxeter algebra, as a free module with coefficients in the xi's.

• When called with a second parameter, say 'y', one specifies that coefficients are in the yi's.

• When this second parameter is 'num' then x1, x2, x3, ... are specialized to 1, 2, 3, ...

• This basis is defined by the recursion : for a simple transposition sk and a permutation perm, such that length(perm sk) > length(perm), then NcaYang(perm sk) = NcaYang(perm) &@* (1 + (x_j - x_i)SgTranspo(k)) where i=perm[k] and j=perm[k+1].

• Coefficients in the expansion are specializations of double Schubert polynomials.

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

EXAMPLES:

> with(NCA):
> NcaYang([3,1,2]);
 
        (x3 - x1) A[2, 1, 3] + (- x3 + x1) (x2 - x3) A[3, 1, 2]
 
               + (x3 - x2) A[1, 3, 2] + A[1, 2, 3]