FUNCTION: SgaYang - compute a special element of the symmetric group algebra

CALLING SEQUENCE:

SgaYang(perm)

SGA[SgaYang](perm)

PARAMETERS:

perm = any list denoting a permutation

SYNOPSIS:

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

• { SgaYang(perm), perm in ListPerm(n) } is a linear basis of the symmetric group 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 SgaYang(perm sk) = SgaYang(perm) &!* (1 + (x_j - x_i) sk) where i=perm[k] and j=perm[k+1].

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

EXAMPLES:

> with(SGA):
> SgaYang([3,1,2]);
 
         (x3 - x2) A[1, 3, 2] - (- x3 + x1) (x3 - x2) A[3, 1, 2]
 
       + A[1, 2, 3] + (x3 - x1) A[2, 1, 3]
 
> SgaYang([3,1,2], 'y');
 
         (y3 - y2) A[1, 3, 2] - (- y3 + y1) (y3 - y2) A[3, 1, 2]
 
       + A[1, 2, 3] + (y3 - y1) A[2, 1, 3]