Let $A_n = \{\sigma \in S_n : \mathrm{sign}\;\sigma=1\}$ be alternating group.
Let $\rho: A_n \to \mathcal{L}(\mathbb{C}^n) $ stand for standart complex representation of $A_n$ defined by $$ \rho(\sigma)(e_i) = e_{\sigma(i)} $$ where $e_i$ is a standart basis of $\mathbb{C}^n$.
It is well known fact that simmilar representation for $S_n$ decomposes to sum of two irreducible ones for subspaces $$ U_1 = \mathbb{C}\sum^n_{i=1} e_i $$ $$ U_0 = \left\{ v \in \mathbb{C}^n \bigg| \sum^n_{i=1} v_i = 0 \right\} $$
But will $U_0$ still be irreducible for $A_n$?
Are there any invariant irreducible subspaces of $U_0$ which exist only for $A_n$?
Clearly $A_2$ is trivial and $A_3$ can't have any irreducible representations of degree 2 as $|A_3| = 3 < 4 = 2^2$. This indicates that there must be one-dimensional invariant subspace of $U_0$ But I don't know how to find equation for this subspace.
You are right, when $n=3$ there is an invariant subspace of $U_0$ of dimension $1$. It is generated by $(1,j,j^2)$.