// Murphy operators

L := function(u,n);
    S := SymmetricGroup(n);
    A := GroupAlgebra(Rationals(),S);
    x := A! 0;
    for v in [1..u-1] do
        x := x + A! S! (v,u);
    end for;
    return x;
end function;

// E_i idempotents in the case of (n-1,1)
//need to know residues of the ith basis element

residue := function(u,i,n); //1 \le i \le n-1, 1 \le u \le n
    if u le i then return u-1; end if;
    if u eq i+1 then return -1; end if;
    return u-2;
end function;

E := function(i,n);
    S := SymmetricGroup(n);
    A := GroupAlgebra(Rationals(),S);
    x := A! 1;
    for c in [-n+1..n-1] do
        for u in [1..n] do
            if not (residue(u,i,n) eq c) then
                x := x * (L(u,n)-c)/(residue(u,i,n)-c);
            end if;
        end for;
    end for;
    return x;
end function;

// Function to compute the Young idempotents for each tableaux
// there should be some relationship between Y(1,n) and E(1,n)
// but it's not obvious what ...

Y := function(i,n);
    S := SymmetricGroup(n);
    A := GroupAlgebra(Rationals(),S);
    C := sub<S | (1, i+1)>;
    Rgens := {};
    for j in [1..i-1] do
        Rgens := Rgens join {S! (j, j+1)};
    end for;
    for j in [i+2,n-1] do
        Rgens := Rgens join {S! (j, j+1)};
    end for;
    R := sub<S | Rgens, (i,i+2)>;
    C; R;
    c := A! 0; for t in C do c := c + Sign(t)*A! t; end for; c;
    r := A! 0; for s in R do r := r + s; end for; r;
    return c*r;
end function;