FUNCTION: HekaJucis - Jucis-Murphy element of the Hecke algebra

CALLING SEQUENCE:

HekaJucis(i)

HEKA[HekaJucis](i)

PARAMETERS:

i = any positive integer

SYNOPSIS:

• The HekaJucis function computes the i-th Jucis-Murphy element inside the Hecke algebra of the symmetric group of degree n (i<=n), where n is taken to be i by default or the second argument if available.

• The algebra generated by the Jucis-Murphy elements is a maximal commutative sub-algebra of the Hecke algebra.

• More explicitly, denoting t(i,j) the transposition exchanging i and j, we have: HekaJucis(1) = 0, HekaJucis(2) = (q1+q2)*t(1,2) - q1*q2, HekaJucis(3) = (q1+q2)t(1,3) - (q1+q2)*q1*q2*t(2,3) + q1^2*q2^2 and HekaJucis(4) = (q1+q2)t(1,4) - (q1+q2)*q1*q2*t(2,4) + (q1+q2)*q1^2*q2^2 t(3,4) - q1^3*q2^3.

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

EXAMPLES:

> with(HEKA):
> HekaJucis(2);
 
                    (q1 + q2) A[2, 1] - q1 q2 A[1, 2]
 
> HekaJucis(4, 6);
 
    (q1 + q2) A[4, 2, 3, 1, 5, 6] - (q1 + q2) q1 q2 A[1, 4, 3, 2, 5, 6]
 
                2   2                         3   3
  + (q1 + q2) q1  q2  A[1, 2, 4, 3, 5, 6] - q1  q2  A[1, 2, 3, 4, 5, 6]