One dimensional SU(n) chains with the same irreducible representation R at each site are considered. We determine which R admit low-energy mappings to a SU (n) /[U (1) ] n - 1 flag manifold sigma model, and calculate the topological angles for such theories. Generically, these models will have fields with both linear and quadratic dispersion relations; for each R, we determine how many fields of each dispersion type there are. Finally, for purely linearly-dispersing theories, we list the irreps that also possess a Zn symmetry that acts transitively on the SU (n) /[U (1) ] n - 1 fields. Such SU(n) chains have an 't Hooft anomaly in certain cases, allowing for a generalisation of Haldane's conjecture to these novel representations. In particular, for even n and for representations whose Young tableaux have two rows, of lengths p1 and p2 satisfying p1 ≠p2, we predict a gapless ground state when p1 +p2 is coprime with n. Otherwise, we predict a gapped ground state that necessarily has spontaneously broken symmetry if p1 +p2 is not a multiple of n.