  Production and verification of multipartite quantum state are an essential
step in quantum information processing. In this work, we propose an efficient
method to decompose symmetric multipartite observables, which are invariant
under permutations between parties, with only $(N+1)(N+2)/2$ local measurement
settings, where $N$ is the number of qubits. We apply the decomposition
technique to evaluate the fidelity between an unknown prepared state and any
target permutation invariant state. In addition, for some typical permutation
invariant states, such as the Dicke state with a constant number of
excitations, $m$, we derive a tight linear bound on the number of local
measurement settings, $m(2m+3)N+1$. Meanwhile, for the $GHZ$ state, the $W$
state, and the Dicke state, we prove a linear lower bound, $\Theta(N)$. Hence,
for these particular states, our decomposition technique is optimal.
