One dimensional SU($n$) chains with the same irreducible representation $mathcal{R}$ at each site are considered. We determine which $mathcal{R}$ admit low-energy mappings to a $text{SU}(n)/[text{U}(1)]^{n-1}$ 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 $mathcal{R}$, we determine how many fields of each dispersion type there are. Finally, for purely linearly-dispersing theories, we list the irreducible representations that also possess a $mathbb{Z}_n$ symmetry that acts transitively on the $text{SU}(n)/[text{U}(1)]^{n-1}$ fields. Such SU($n$) chains have an t Hooft anomaly in certain cases, allowing for a generalization of Haldanes conjecture to these novel representations. In particular, for even $n$ and for representations whose Young tableaux have two rows, of lengths $p_1$ and $p_2$ satisfying $p_1 ot=p_2$, we predict a gapless ground state when $p_1+p_2$ is coprime with $n$. Otherwise, we predict a gapped ground state that necessarily has spontaneously broken symmetry if $p_1+p_2$ is not a multiple of $n$.
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