FUNCTION: BnIdcaYang - Yang-Baxter element of the Bn-idCoxeter algebra

CALLING SEQUENCE:

BnIdcaYang(code)

BNA[BnIdcaYang](code)

PARAMETERS:

code = any list denoting a code

SYNOPSIS:

• The BnIdcaYang function calculates a Yang-Baxter element of the Bn-idCoxeter algebra.

• The set BnIdcaYang(code), for all codes in Bn is a linear basis of the Bn-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 reflection sk, k>0 and a code of B(n), such that length(code sk)>length(code), then one has BnIdcaYang(code sk) = BnIdcaYang(code) &!$* (1 + (1 - (x_j1*x_j2) / (x_i1*x_i2)) Pk) where i1=palin[n-k], i2=palin[n+k], j1=palin[n-k+1], j2=palin[n+k-1] and palin is the BnCode2Palin(code). If k=0, then the expression becomes BnIdcaYang(code sk) = BnIdcaYang(code) &!$* (1 + (1 - (x_j/x_i)) P0) where i=palin[n] and j=palin[n+1]. P0, P1, P2, ... denote the generators of the Bn-idCoxeter algebra.

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

EXAMPLES:

> with(BNA):
> BnIdcaYang([0,2]);
 
     (- x1 x3 + x2 x4) B[0, 1]   (- x1 + x4) (- x1 x3 + x2 x4) B[0, 2]
   - ------------------------- + -------------------------------------
               x1 x3                               2
                                                 x1  x3
 
          (- x1 + x4) B[1, 0]
        - ------------------- + B[0, 0]
                   x1
 
> BnIdcaYang([0,2], 'num');
 
         2                    3          2                  3
   (1 - q ) B[0, 1] + (- 1 + q ) (- 1 + q ) B[0, 2] + (1 - q ) B[1, 0]
 
   + B[0, 0]