FUNCTION: IdcaYang - compute a special element of the idCoxeter algebra

CALLING SEQUENCE:

IdcaYang(perm)

IDCA[IdcaYang](perm)

PARAMETERS:

perm = any list denoting a permutation

SYNOPSIS:

• The IdcaYang function calculates a special element of the idCoxeter algebra.

• { IdcaYang(perm), perm in ListPerm(n) } is a linear basis of the idCoxeter 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 q^1, q^2, q^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 IdcaYang(perm sk) = IdcaYang(perm) &$* (1 + (1 - x_j / x_i) sk) where i=perm[k] and j=perm[k+1].

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

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

EXAMPLES:

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