  Color structures for tree level scattering amplitudes in gauge theory are
studied in order to determine the symmetry properties of the color-ordered
sub-amplitudes. We mathematically formulate the space of color structures
together with the action of permuting external legs. The character generating
functions are presented from the mathematical literature and we determine the
decomposition into irreducible representations. Mathematically, free Lie
algebras and the Lie operad are central. A study of the implications for
sub-amplitudes is initiated and we prove directly that both the Parke-Taylor
amplitudes and Cachazo-He-Yuan amplitudes satisfy the Kleiss-Kuijf relations.
