FUNCTION: Sga2Carre - Carre element of the symmetric group algebra

CALLING SEQUENCE:

Sga2Carre(e)

SGA[Sga2Carre](e)

PARAMETERS:

e = any element of the symmetric group algebra

SYNOPSIS:

• The Sga2Carre function converts an element of the symmetric group algebra into a Carre operator by replacing each A[perm] by the corresponding Carre element of the symmetric group algebra.

• The Sga2Carre together with the Sga2Nabla functions are the building blocks of idempotents in the symmetric group algebra.

• Given a permutation perm, Sga2Carre(A[perm]) is a certain product of factors of the type (SgTranspo(i,n) + 1/k) where the shifts 1/k are given by Yang-Baxter conditions. Special cases provide idempotents. For example, Sga2Carre(A[n, n-1, ..., 2, 1]) = sum of all permutations of Sn.

• They are defined by the recursion: for a simple transposition si and a permutation perm, such that length(perm si) > length(perm), then Sga2Carre(A[perm si]) = Sga2Carre(A[perm]) &!* (si + 1/k), in which k=perm[i+1] - perm[i].

• Whenever there is a conflict between the function name Sga2Carre and another name used in the same session, use the long form SGA['Sga2Carre'].

EXAMPLES:

> with(SGA):
> Sga2Carre(q*A[2,3,1] + A[1,2,3]);
 
  q A[2, 1, 3] + (1/2 q + 1) A[1, 2, 3] + 1/2 q A[1, 3, 2] + A[2, 3, 1] q