  In this work we consider the weakly coupled Schr\"odinger cubic system \[
\begin{cases} \displaystyle -\Delta u_i+\lambda_i u_i= \mu_i u_i^{3}+
u_i\sum_{j\neq i}b_{ij} u_j^2 \\ u_i\in H^1(\mathbb{R}^N;\mathbb{R}), \quad
i=1,\ldots, d, \end{cases} \] where $1\leq N\leq 3$, $\lambda_i,\mu_i >0$ and
$b_{ij}=b_{ji}>0$ for $i\neq j$. This system admits semitrivial solutions, that
is solutions $\mathbf{u}=(u_1,\ldots, u_d)$ with null components. We provide
optimal qualitative conditions on the parameters $\lambda_i,\mu_i$ and $b_{ij}$
under which the ground state solutions have all components nontrivial, or,
conversely, are semitrivial.
  This question had been clarified only in the $d=2$ equations case. For $d\geq
3$ equations, prior to the present paper, only very restrictive results were
known, namely when the above system was a small perturbation of the
super-symmetrical case $\lambda_i\equiv \lambda$ and $b_{ij}\equiv b$. We treat
the general case, uncovering in particular a much more complex and richer
structure with respect to the $d=2$ case.
