Symmetric-group decomposition of SU(N) group-theory constraints on four-, five-, and six-point color-ordered amplitudes at all loop orders

Color-ordered amplitudes for the scattering of n particles in the adjoint representation of SU(N ) gauge theory satisfy constraints that arise from group theory alone. These constraints break into subsets associated with irreducible representations of the symmetric group Sn, which allows them to be presented in a compact and natural way. Using an iterative approach, we derive the constraints for six-point amplitudes at all loop orders, extending earlier results for n = 4 and n = 5. We then decompose the four-, five-, and six-point group-theory constraints into their irreducible Sn subspaces. We comment briefly on higher-point two-loop amplitudes.