FUNCTION: HekaYang - compute a special element of the Hecke algebra

CALLING SEQUENCE:

HekaYang(perm)

HEKA[HekaYang](perm)

PARAMETERS:

perm = any list denoting a permutation

SYNOPSIS:

• The HekaYang function calculates a special element of the Hecke algebra of the symmetric group.

• { HekaYang(perm), perm in ListPerm(n) } is a linear basis of the Hecke algebra of the symmetric group, as a free module with coefficients in the xi's, q1, and q2.

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

• This basis is defined by the recursion : for a simple transposition sk and a permutation perm, such that length(perm sk) > length(perm), then HekaYang(perm sk) = HekaYang(perm) &?* (1 + (x_j / x_i - 1)/(q1+q2) sk) where i=perm[k] and j=perm[k+1].

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

EXAMPLES:

> with(HEKA):
> HekaYang([2,3,1]);
 
                        (- x2 + x1) A[2, 1, 3]
                      - ---------------------- + A[1, 2, 3]
                             x1 (q1 + q2)
 
                                   2
      (- x3 x2 + x1 x2 + x3 x1 - x1 ) A[2, 3, 1]   (x3 - x1) A[1, 3, 2]
    - ------------------------------------------ + --------------------
                             2   2                     x1 (q1 + q2)
                    (q1 + q2)  x1