FUNCTION: BnGaYang - Yang-Baxter element of the Bn-group algebra

CALLING SEQUENCE:

BnGaYang(code)

BNA[BnGaYang](code)

PARAMETERS:

code = any list denoting a code

SYNOPSIS:

• The BnGaYang function calculates a Yang-Baxter element of the Bn-group algebra.

• The set of BnGaYang(code), for all codes in Bn, is a linear basis of the Bn-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 reflection sk, k>0 and a code, such that length(code sk) > length(code), then one has BnGaYang(code sk) = BnGaYang(code) &!!* (1 + (sign(perm[k])*x_j - sign(perm[k+1])*x_i) sk) where i=abs(perm[k]), j=abs(perm[k+1]) and perm is the corresponding signed permutation. When k=0, one has: BnGaYang(code s0) = BnGaYang(code) &!!* (1 - 2*(sign(perm[1])*x_i) s0) in which i=abs(perm[1]).

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

EXAMPLES:

> with(BNA):
> BnGaYang([0,2]);
 
    (x2 - x1) B[0, 1] + B[0, 0] - 2 (x2 - x1) x2 B[0, 2] - 2 x2 B[1, 0]
 
> BnGaYang([0,2], 'num');
 
                   B[0, 1] + B[0, 0] - 4 B[0, 2] - 4 B[1, 0]