FUNCTION: SgaJucis - Jucis-Murphy element of the symmetric group algebra

CALLING SEQUENCE:

SgaJucis(i)

SGA[SgaJucis](i)

PARAMETERS:

i = any positive integer

SYNOPSIS:

• The SgaJucis function computes the i-th Jucis-Murphy element inside the 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 symmetric group algebra.

• More explicitly, denoting T(i,j) the transposition exchanging i and j, we have: SgaJucis(1) = 0, SgaJucis(2) = T(1,2), SgaJucis(3) = T(1,3) + T(2,3), ..., SgaJucis(k) = T(1,k) + T(2,k) + ... + T(k-1,k).

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

EXAMPLES:

> with(SGA):
> SgaJucis(2);
 
                                   A[2, 1]
 
> SgaJucis(4, 6);
 
       A[4, 2, 3, 1, 5, 6] + A[1, 4, 3, 2, 5, 6] + A[1, 2, 4, 3, 5, 6]