Considering the representation of $S_n$ where the objects being permuted are the basis vectors of an $n$ dimensional vector space
$$ |1 \rangle, |2 \rangle, \:... \:, |n \rangle$$
If the representation operator $D$ takes $|j \rangle$ to $|k \rangle$, the matrix elements of this representation are
$$\langle l|D|j \rangle = \delta_{kl}$$
I started finding the projection operator into the trivial representation but I don't know how to proceed from there/if it is the right path. I am just starting to learn group theory from the book H. Georgi - Lie Algebra In Particle Physics and any help would be appreciated. Tahnks.
Hint If we denote $$\Bbb V := \operatorname{span}\{|1\rangle, \ldots. |n\rangle\},$$ the representation is a map $S_n \mapsto \operatorname{End} \Bbb V$. Suppose the action of $S_n$ fixes $\sum_{i = 1}^n a_i | i \rangle$. What can we say about the coefficients $a_i$?