Why does the action of $S_3$ representation not equal to the direct sum of its irreducible representations

Question

This might be a really simple question..

The representation $(\rho,\mathbb C^3)$ of $S_3$ can be decomposed to $(\rho_1\oplus \rho_3,W\oplus W')$, where $W=\{k(1,1,1)|k\in \mathbb C\}$, $W'=\{(x_1,x_2,x_3)|x_1+x_2+x_3=0\}$,
$\rho_1(\sigma)=1\forall \sigma \in S_3$ and $\rho_3$ is the standard representation depicted in the following link: https://groupprops.subwiki.org/wiki/Standard_representation_of_symmetric_group:S3.

So, for instance, why is it that $\rho(123)(x_1,x_2,x_3)=(x_3,x_1,x_2)\neq (x_1,-x_3,x_2-x_3)=(\rho_1\oplus \rho_3)(123)(x_1,x_2,x_3)$?