Find a certain decomposition of $\mathbb{C}S_3$

Question

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 :)

Answer 1

My solution:

Following the explanation in the comments we get:

Choosing the canonical basis $\mathcal{B}=S_3$

$$\mathbb{C}S_3e_1=\mathbb{C}e_1\Longrightarrow \mathbb{C}(1,1,1,1,1,1)^T$$

$$\mathbb{C}S_3e_2=\mathbb{C}e_2\Longrightarrow \mathbb{C}(1,-1,-1,-1,1,1)^T$$

$\mathbb{C}S_3e_3 $ is the orthogonal complement of the span two other vectors combined.So:

$$\mathbb{C}S_3e_3=\text{Span}((1,0,0,0,0,-1)^T,\,(1,0,0,0,-1,0)^T),\,(0,1,-1,0,0,0)^T, (0,1,0,-1,0,0)^T)$$