Collective symmetry algebras of the periodic Fredkin chain

We determine the nonlocal symmetry algebra of the periodic spin-1/2 Fredkin chain for every length N ≥ 3. A rank-one generating function shows that the conserved raising and lowering charges act only on the symmetric Dicke multiplet and vanish on its orthogonal complement. An auxiliary spin on operator space yields a closed formula for their collective-spin expansion. The generated algebra is symplectic at odd length and orthogonal at even length, correcting the type labels in the original conjecture while confirming its rank and central splitting. For ideal independent charge controls, we characterize pure-state orbits and optimal transfer fidelities. At even length, we prove the path-sign symmetry and determine how additional controls act on the full ground space. We also obtain the operator Schmidt spectrum of every charge and an exact matrix product representation with minimal bond dimension at every cut.