Permutation-symmetric three-body O(6) hyperspherical harmonics in three spatial dimensions

We have constructed the three-body permutation symmetric O(6) hyperspherical harmonics which can be used to solve the non-relativistic three-body Schr{\" o}dinger equation in three spatial dimensions. We label the states with eigenvalues of the $U(1) \otimes SO(3)_{rot} \subset U(3) \subset O(6)$ chain of algebras and we present the corresponding $K \leq 4$ harmonics. Concrete transformation properties of the harmonics are discussed in some detail.