I need to find a decomposition of $\mathbb{C}S_3$ in the following way:
$\mathbb{C}S_3=\mathbb{C}S_3e_1\oplus\mathbb{C}S_3e_2\oplus\mathbb{C}S_3e_3$
with $e_i=|G|^{-1}\sum\limits_{g\in G}\chi_i(\text{id})\chi_i(g^{-1})g$ and $\chi_1,\chi_2,\chi_3$ being the irreduceable characters.
$\chi_1 $ being the trivial character, $\chi_2$ being the singum function and $\chi_3: \text{id}\mapsto 2,\, [(12)]\mapsto 0, \, [(123)]\mapsto -1$
I calculated:
$e_1=1/6\,( \,\text{id}+(1 2) +(13)+(23)+(123)+(132)\,)$
$e_2=1/6\,( \,\text{id}-(1 2) -(13)-(23)+(123)+(132)\,)$
$e_3=1/6\,( \,4\text{id}-2(123)-2(132)\,)$
But from here I dont know how to proceed.. could someone explain to me how I can use my $e_i$ to get the decomposition?
thanks :)