Consider the group $S_n$ and the action on a linearized set of elements $\{v_1, \dots, v_n\}$ $V= \{\sum_{i=1}^nc_iv_i\}$ (generated vector space).
I want to show that the representation
$$\varphi: S_n \rightarrow GL(V)$$
has exactly two proper (non-zero) subrepresentations and $V$ is their direct sum.
Now I know that if we can write $V = V_1 \oplus V_2$ and $V$ is carrying a linear representation, then there exist $\varphi_1 \, , \, \varphi_2$ such that (with a little abuse of notation) , $\varphi_g = \varphi^{1}_g \oplus \varphi^2_g $ and $\varphi_g$ will be a block-diagonal matrix
$$\begin{pmatrix}\varphi^1_g(v^{(1)}_1, \dots v^{(1)}_n) & 0 \\ 0 & \varphi^2_g(v^{(2)}_1, \dots v^{(2)}_n)\end{pmatrix}$$
But how can I find the proper subrepresentation in this case?
I would say that expressing $V$ as a direct sum in this case would yield just $V = V \oplus \{0\}$. But in this case I would have that one of subrepresentations is zero.