Subrepresentations as subspaces of $V$ Let $V = \{ S \subset \{1,2,3,4\} : \vert S \vert = 2\}$ which we think of as the set of the edges of the tetrahedron, where an edge is labelled by the two vertices which it connects. For example, the edge connecting the vertices labelled $1$ and $2$ is named $(12)$.

Now let $\rho$ be the permutation representation acting on $V$. By comparing the character of $\rho$, $X_\rho$, to the character table of $S_4$ one can find that: $$X_\rho = 1 + X + \varphi$$ Where 1 is the character of the trivial representation,  $X$ is the character of one of the 3-D representations of $S_4$ and $\varphi$ is the character of the unique 2D representation of $S_4$. As far as I understand, this implies that $V \cong W_0 \oplus W_1 \oplus W_2$, where these $W_i$-spaces are the subspaces of $V$ which are invariant under these representation (at least up to isomorphism), I'd like to find these explicitly as subspaces of $V$. However a few things confuse me. Firstly, should I expect these subspaces to be subrepresentations of the permutation representation? If not, how can I hope to find them as subspaces of $V$?
I think I can see that if we consider the trivial representation, then any edge will generate the 1d space... I have no idea for the other two spaces.
Any help would be massively appreciated.
 A: Let's first list the standard basis of $V$
$$
e_1 = (1,2)\\
e_2 = (1,3)\\
e_3 = (1,4)\\
e_4 = (2,3)\\
e_5 = (2,4)\\
e_6 = (3,4)\\
$$
For each of the irreducible summands, lets call them $a$, $b$ and $c$ and subscripts to say the basis of each.
So you say you already have the trivial representation
$$
a_1 = e_1 + e_2 + e_3 + e_4 + e_5 + e_6
$$
You can see this is preserved by $S_4$ because it just permutes the summands. So that's the subspace $W_0 = \mathbb{C} a_1$ taken care of.
Now for the other two. We need to think of other phenomena that are preserved by the $S_4$ action. One of them is if two edges are opposite on the tetrahedron, then no matter what you do with $S_4$, they will still be opposite. So use that idea to write some more vectors in $V$.
$$
b_1 = e_1 - e_6\\
b_2 = e_2 - e_5\\
b_3 = e_3 - e_4\\
$$
The $S_4$ action permutes and changes signs on these, but leaves $W_1 = \bigoplus \mathbb{C} b_i$ invariant as a subspace.
I'll leave the basis and how they are interpreted as edges for the last 2 dimensions up to you.
