FUNCTION: StdTab2SgaIdemp - build an idempotent from a standard tableau

CALLING SEQUENCE:

StdTab2SgaIdemp(t)

TAB[StdTab2SgaIdemp](t)

PARAMETERS:

t = any list of list(s) denoting a standard tableau

SYNOPSIS:

• The StdTab2SgaIdemp function computes an idempotent of the symmetric group algebra from a standard tableau. These idempotents (for all standard tableaux) are mutually orthogonal.

• Let t' be the tableau obtained from t by deleting the last letter n and let t1, t2, ..., be the other tableaux obtained from t' by adding n. Then, up to a normalization factor: StdTab2SgaIdemp(t)=StdTab2SgaIdemp(t')*product(SgaJucis(n,n)-c(n,ti)) where SgaJucis(n,n) is the last Jucis-Murphy element and c(n,ti) the distance to the diagonal of the letter n in the tableau ti.

• Whenever there is a conflict between the function name StdTab2SgaIdemp and another name used in the same session, use the long form TAB['StdTab2SgaIdemp'].

EXAMPLES:

> with(TAB):
> r1:=StdTab2SgaIdemp([[3], [1,2]]):
> r2:=StdTab2SgaIdemp([[3], [1,2,4]]);
 
      r2 := 5/48 A[4, 2, 3, 1] + 5/48 A[1, 4, 3, 2] + 1/24 A[1, 2, 4, 3]
 
         + 1/8 A[1, 2, 3, 4] + 5/48 A[2, 4, 3, 1] - 1/48 A[3, 2, 4, 1]
 
         + 5/48 A[4, 1, 3, 2] - 1/48 A[1, 3, 4, 2] - 1/48 A[4, 2, 1, 3]
 
         - 1/48 A[1, 4, 2, 3] - 1/48 A[2, 3, 4, 1] - 1/12 A[3, 4, 2, 1]
 
         - 1/48 A[2, 4, 1, 3] - 1/12 A[4, 3, 1, 2] - 1/16 A[3, 2, 1, 4]
 
         - 1/16 A[1, 3, 2, 4] - 1/12 A[4, 3, 2, 1] - 1/48 A[3, 1, 4, 2]
 
         - 1/12 A[3, 4, 1, 2] - 1/48 A[4, 1, 2, 3] - 1/16 A[2, 3, 1, 4]
 
         - 1/16 A[3, 1, 2, 4] + 1/8 A[2, 1, 3, 4] + 1/24 A[2, 1, 4, 3]
 
> (r1 &!* (SgaJucis(4,4) &!* (SgaJucis(4,4) + 2) / 8)) &!- r2;
 
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